HSG.SRT.C.6: Similar Right Triangles and the Definitions of Sine, Cosine and Tangent
In plain English: HSG.SRT.C.6 is the Common Core geometry standard that asks students to understand why side ratios in right triangles depend only on the angles. Because right triangles with an equal acute angle are similar, ratios such as opposite leg over hypotenuse stay fixed, and this is how sine, cosine and tangent of an acute angle are defined. It is usually taught in Geometry.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Common Core State Standards for Mathematics · Domain: Similarity, Right Triangles, and Trigonometry (SRT) · Cluster: Define trigonometric ratios and solve problems involving right triangles Also written as HSG-SRT.C.6 or G-SRT.6 · Official standard
Students discover and then justify that every right triangle with a given acute angle has the same side ratios. The justification is similarity: two right triangles with one pair of congruent acute angles are similar by AA, so their corresponding sides are proportional, and any ratio of two sides inside one triangle equals the matching ratio in the other.
Because the ratio belongs to the angle and not to a particular triangle, it can be named after the angle. That is the definition of sine, cosine and tangent of an acute angle. Students name the sides relative to each acute angle, compute exact ratios from side lengths, measure drawn triangles, and compare their results with calculator values.
Learning Objectives
By the end of this lesson, students will be able to:
Explain, using AA similarity, why two right triangles with a congruent acute angle have equal side ratios
Identify the opposite leg, adjacent leg and hypotenuse relative to either acute angle of a right triangle
Define sin A, cos A and tan A for an acute angle A as ratios of side lengths
Compute exact trigonometric ratios from given side lengths and explain why a calculator value matches a measured triangle
Use a known ratio of an angle to find a side in a larger or smaller similar right triangle
Prior Knowledge Required
Students should already be comfortable with:
The AA criterion for similar triangles HSG.SRT.A.3
Similarity and proportional corresponding sides HSG.SRT.A.2
The Pythagorean Theorem for finding a missing side 8.G.B.7
Writing and simplifying ratios and fractions
Measuring lengths with a ruler and angles with a protractor
Sketch four right triangles on the board, each with its legs and hypotenuse labeled, and mark the angle at the left end of the longer leg in each one. The side lengths are 3-4-5, 6-8-10, 9-12-15 and 5-12-13.
Warm-Up Prompt
"For each triangle, divide the side across from the marked angle by the hypotenuse. Which triangles give the same answer? What do you think those triangles have in common?"
The first three triangles give 3/5, 6/10 and 9/15, which are all 0.6. The last one gives 5/13 ≈ 0.385. Ask students to check whether the first three are similar (each is a scale copy of 3-4-5) and whether the fourth is similar to them (it is not: 5/3, 12/4 and 13/5 are not equal). Record the conjecture: "Right triangles that are the same shape give the same ratio." The lesson turns that conjecture into an argument.
Direct Instruction20 minutes
Part 1: Why the ratio is a property of the angle. Use Diagram 1. Every right triangle that has an acute angle equal to angle A also has a 90° angle, so any two of them are similar by the AA criterion (HSG.SRT.A.3). Walk through the argument for two right triangles ABC and DEF with right angles at C and F and ∠A ≅ ∠D:
Two pairs of congruent angles: ∠C ≅ ∠F (both right) and ∠A ≅ ∠D (given), so △ABC ~ △DEF by AA.
Corresponding sides are proportional: BC/EF = AB/DE = AC/DF = k, the scale factor.
Rearrange one proportion: from BC/EF = AB/DE, multiply both sides by EF/AB to get BC/AB = EF/DE.
Read the result: the leg opposite the angle divided by the hypotenuse is the same number in both triangles. The same steps work for the other two ratios.
Conclusion: the ratio depends only on the measure of the acute angle, so it is fair to give it a name that belongs to the angle.
Part 2: The definitions. Use Diagram 2. For an acute angle A in a right triangle, name the sides relative to A: the leg opposite A, the leg adjacent to A (the leg that is a side of the angle), and the hypotenuse. Define sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse and tan A = opposite/adjacent. Stress that "opposite" and "adjacent" change when you switch to the other acute angle, while the hypotenuse stays the same.
Same angle, two sizes
Right triangles ABC and DEF have right angles at C and F and ∠A ≅ ∠D. In △ABC, BC = 8, AC = 15, AB = 17. In △DEF, EF = 24, DF = 45, DE = 51.
Equation: sin A = 8/17 and sin D = 24/51 = 8/17: the scale factor 3 cancels
All three ratios
In right △ABC with right angle C, BC = 20, AC = 21 and AB = 29 (Diagram 2). Name the ratios for ∠A.
Equation: sin A = 20/29, cos A = 21/29, tan A = 20/21
The other acute angle
Use the same triangle, but stand at ∠B. Now AC is opposite and BC is adjacent.
Equation: sin B = 21/29, cos B = 20/29, tan B = 21/20
Using a fixed ratio
A small right triangle shows that tan P = 3/4 for an angle P. A larger right triangle has the same angle P, and the leg adjacent to P is 28 cm. How long is the opposite leg?
Equation: opposite = (3/4)(28) = 21 cm
Measured triangle and calculator
A student draws a right triangle with a 28° angle and a 10 cm hypotenuse and measures the opposite leg.
Equation: about 4.7 cm, so the ratio is about 0.47; a calculator gives sin 28° ≈ 0.4695
After the last example, point out that the calculator does not know anything about the student's drawing: it returns the same ratio because every right triangle with a 28° angle is similar to that one. Check that calculators are in degree mode.
Guided Practice15 minutes
Pairs work through three triangles. For each one they find the three ratios for the marked angle, then compare with the pair next to them.
Guided practice triangles
Triangle
Opposite
Adjacent
Hypotenuse
sin, cos, tan of the marked angle
1
7
24
25
7/25, 24/25, 7/24
2
14
48
50
14/50 = 7/25, 48/50 = 24/25, 14/48 = 7/24
3
9
40
41
9/41, 40/41, 9/40
Ask: "Triangles 1 and 2 gave the same ratios. What must be true about their marked angles?" (They are congruent, because the triangles are similar.) Listen for students who use the hypotenuse as the adjacent side, and for students who flip a ratio when the marked angle is at the top of the triangle instead of the left.
Independent Practice15 minutes
Students work alone on three tasks:
A right triangle has legs 12 and 35 and hypotenuse 37. Write all three ratios for both acute angles, six values in all.
A 30-60-90 triangle has a short leg of 4. Find the other two sides (8 and 4√3), then write sin 30°, cos 30° and tan 30° as exact values (1/2, √3/2 and √3/3).
Explain in two sentences why a different 30-60-90 triangle, with a short leg of 7, gives the same three values.
Students check the first task by confirming that the sine of one acute angle equals the cosine of the other, and they check the second with a calculator.
Closure5-10 minutes
Exit ticket: (1) Right triangles PQR and XYZ each have a 52° angle. Explain in one or two sentences why the leg opposite the 52° angle divided by the hypotenuse is the same number in both. (2) A right triangle has legs 16 and 63. Find the hypotenuse, then write sin and cos of the angle opposite the 16-unit leg. (Answer: 65; 16/65 and 63/65.)
Differentiation Strategies
For Struggling Students
Give a template with the three side names printed in the triangle, and have students first highlight the marked angle and trace the side across from it
Start with triangles drawn in the same orientation, then rotate them once students label sides correctly
Provide a proportion frame for the similarity argument: "BC/EF = AB/DE, so BC/AB = ____ / ____"
For Advanced Students
Ask students to prove that (sin A)² + (cos A)² = 1 for any acute angle A using the Pythagorean Theorem
Ask why tan A can be greater than 1 while sin A and cos A cannot, and for which angles tan A is greater than 1
Have students draw a line through the origin at a given angle to the x-axis and explain why its slope equals the tangent of that angle
Assessment Guidance
What to Look For
Listen for the word "similar" in students' explanations: the standard asks them to understand why the ratio depends only on the angle, not only to compute it. Check that students name opposite and adjacent relative to the angle in the question, especially when the angle is not at the left of the drawing. When students compute a ratio from a measured triangle, expect small differences from the calculator value and ask them to explain where those differences come from.
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Classroom Activities
3 Activities
1
Same Angle, Different Sizes
20 minGroups of 4
Each group studies one angle. Every student draws a right triangle with that angle but a different hypotenuse, measures the sides, and computes the three ratios. The group compares results and then compares them with calculator values.
Procedure
Assign each group one angle: 20°, 35°, 55° or 70°
Each student draws a right triangle with the assigned angle and a hypotenuse of 6, 9, 12 or 15 cm (one length per student)
Students measure the two legs to the nearest millimeter and compute opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent to two decimal places
The group records all four sets of ratios on a poster, then adds the calculator values for the angle
Expected Values
20°: sin ≈ 0.34, cos ≈ 0.94, tan ≈ 0.36
35°: sin ≈ 0.57, cos ≈ 0.82, tan ≈ 0.70
55°: sin ≈ 0.82, cos ≈ 0.57, tan ≈ 1.43
70°: sin ≈ 0.94, cos ≈ 0.34, tan ≈ 2.75
Discussion Questions
Why are the four ratios in your group close to each other even though the triangles are different sizes?
Why are they not exactly equal?
Compare the 35° and 55° posters. What do you notice about the sine and cosine values?
Modification for Distance Learning
Students build the triangles in a free dynamic geometry tool, drag a vertex to change the size while the angle stays fixed, and record the ratios at three sizes.
2
Triangle Family Sort
20 minPairs
Pairs receive 12 cards, each showing a right triangle by its three side lengths, with the angle opposite the shortest leg marked. Pairs sort the cards into families that share the same marked angle, using ratios instead of a protractor.
Card Set
Family 1: 3-4-5, 6-8-10, 1.5-2-2.5
Family 2: 5-12-13, 10-24-26, 15-36-39
Family 3: 8-15-17, 16-30-34, 4-7.5-8.5
Family 4: 7-24-25, 14-48-50, 21-72-75
Shuffle the 12 cards before handing them out.
Procedure
For each card, compute the tangent of the marked angle (shortest leg divided by longer leg) as a simplified fraction
Group cards with equal tangents, then confirm with the sine
For one family, write the scale factor between each pair of cards
Write one sentence explaining why equal ratios mean equal angles
Discussion Questions
Could two cards have the same sine for the marked angle but a different tangent? Why not?
Why is it enough to check one ratio to know the marked angles are congruent?
Challenge Variation
Each pair creates a fifth family of three cards, including one card with a non-integer side, and trades it with another pair to sort.
3
Write the Similarity Argument
20 minPairs, then whole class
Pairs write a short, complete argument that the cosine of an angle does not depend on the triangle used to compute it. The class then compares arguments and builds one model version.
Prompt
"Right triangles GHK and MNP have right angles at K and P, and ∠G ≅ ∠M. Show that GK/GH = MP/MN." Students must name the similarity criterion, write the proportion, and show the algebra step that rearranges it.
Procedure
Partner A draws and labels both triangles, marking the congruent angles
Partner B writes the argument in numbered statements with a reason for each
Pairs swap with another pair and check: is the criterion named, and is the rearranging step shown?
Two pairs present; the class agrees on a model argument and adds a final line: "So cos G = cos M."
Discussion Questions
Why do we only need one pair of acute angles to be congruent?
Would the argument work for two triangles that are not right triangles? What would the ratios mean?
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Diagrams & Visual Aids
2 diagrams
Diagram 1: Similar Right Triangles Give the Same Ratios
Three right triangles drawn to scale share the acute angle A, so they are similar by AA. Their sides are 3-4-5, 6-8-10 and 9-12-15, and in each one the leg opposite A divided by the hypotenuse is 0.6. The ratio is a property of angle A.
Diagram 2: Naming the Sides Relative to Angle A
A 20-21-29 right triangle drawn to scale, with the right angle at C. Relative to angle A, BC is the opposite leg, AC is the adjacent leg and AB is the hypotenuse. The three ratios of these sides define sin A, cos A and tan A.
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Homework Assignment
~30 min
HSG.SRT.C.6 Homework: Similarity and Trigonometric Ratios
Directions: Show all work. Give exact ratios as simplified fractions unless a problem asks for a decimal. When a problem asks you to explain, name the similarity criterion you use.
Part 1: Why the Ratios Are Fixed (Problems 1-3)
Right triangles JKL and MNP have right angles at L and P, and m∠J = m∠M = 62°. Explain why KL/JK = NP/MN. Which trigonometric ratio of 62° does this show is the same in both triangles?
In right △RST, the right angle is at T, ST = 11 and RT = 60. Find RS, then write sin R, cos R and tan R.
Right triangle R′S′T′ has sides 33, 180 and 183, and ∠R′ ≅ ∠R from Problem 2. A student says, "sin R′ = 33/183, so it is a different number from sin R." Is the student right? Explain using both the numbers and similarity.
Part 2: Using the Definitions (Problems 4-5)
For an acute angle θ in a right triangle, cos θ = 28/53. Sketch a right triangle that fits, find the missing side, and write sin θ and tan θ.
Draw a right triangle with a 40° angle and an 8 cm hypotenuse. Measure both legs, compute sin 40°, cos 40° and tan 40° from your measurements to two decimal places, and compare them with calculator values. Explain any difference.
Part 3: Applying the Idea (Problem 6)
Two wheelchair ramps rise at the same angle. The first ramp rises 1 foot over a horizontal run of 12 feet. The second has a horizontal run of 30 feet. How high does the second ramp rise? Explain why the tangent of the ramp angle lets you answer without measuring the angle.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Similarity Reasoning
Names AA, writes the proportion and rearranges it correctly
Mentions similarity but skips the proportion step
No reason given
Side Names
Opposite, adjacent and hypotenuse correct for every angle used
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
In right triangle DEF, ∠E is the right angle. Which ratio is cos F?
Answer: B
The leg adjacent to ∠F is EF (it runs from F to the right-angle vertex E), and the hypotenuse is DF, so cos F = EF/DF. Choice A is opposite over hypotenuse, which is sin F. Choice C is opposite over adjacent, which is tan F.
Question 2 of 20 · Multiple Choice
Why does every right triangle with a 25° acute angle give the same value of (leg opposite the 25° angle)/(hypotenuse)?
Answer: D
Each triangle has a 90° angle and a 25° angle, so any two of them are similar by AA, and ratios of corresponding sides are equal. Choice A is the error of confusing similar with congruent: the triangles can be different sizes. Choice C is wrong because the Pythagorean Theorem does not fix the size.
Question 3 of 20 · Multiple Choice
A right triangle has legs 33 and 56 and hypotenuse 65. What is the sine of the angle opposite the 33-unit leg?
Answer: A
Sine is opposite over hypotenuse: 33/65. Choice B is the cosine of that angle (adjacent over hypotenuse). Choice C is the tangent, and choice D is the reciprocal of the sine.
Question 4 of 20 · Multiple Choice
A right triangle has sides 13, 84 and 85. Angle θ is opposite the 13-unit leg. What is cos θ?
Answer: C
The leg adjacent to θ is 84 and the hypotenuse is 85, so cos θ = 84/85. Choice A uses the opposite leg, which gives sin θ. Choice B is tan θ.
Question 5 of 20 · Multiple Choice
A right triangle has legs 36 and 77 and hypotenuse 85. What is the tangent of the angle opposite the 36-unit leg?
Answer: B
Tangent is opposite over adjacent: 36/77. Choice C flips the ratio, which is the tangent of the other acute angle. Choice A is the sine.
Question 6 of 20 · Multiple Choice
Right triangle PQR has its right angle at R, with QR = 20, PR = 99 and PQ = 101. Right triangle STU has its right angle at U, ∠S ≅ ∠P, and ST = 202. What is sin S?
Answer: A
The triangles are similar by AA, so sin S = sin P = QR/PQ = 20/101. The larger triangle has UT = 40 and ST = 202, and 40/202 is the same fraction. Choice C mixes a side of the small triangle with a side of the large one. Choice B is cos P.
Question 7 of 20 · Multiple Choice
For an acute angle θ, a small right triangle shows that sin θ = 0.8. A larger right triangle with the same angle θ has a hypotenuse of 45 cm. How long is the leg opposite θ?
Answer: D
Because sin θ is the same in every right triangle with angle θ, opposite/45 = 0.8, so the opposite leg is 0.8(45) = 36 cm. Choice A divides 45 by 0.8 instead of multiplying. Choice B uses the cosine, 0.6, which is the adjacent leg.
Question 8 of 20 · Multiple Choice
Right triangle JKL has its right angle at L, with KL = 5 (opposite ∠J) and JL = 12, so tan J = 5/12. A new right triangle keeps the 12-unit leg next to the angle at J but has an opposite leg of 10. What happens to the tangent of the angle at J?
Answer: B
Only one leg changed, so the legs are not in the same ratio (10/5 is not 12/12) and the new triangle is not a scale copy of JKL. Its angle at J is a different angle, with tangent 10/12 = 5/6. Choice A applies the same-shape rule to a triangle whose shape changed. Choice C doubles the adjacent leg instead of the opposite leg. Choice D has the right tangent but assumes the angle doubles: tan J = 5/12 gives J ≈ 22.6°, and tan J = 5/6 gives about 39.8°, not 45.2°.
Question 9 of 20 · Multiple Choice
In right triangle XYZ, the right angle is at Y. Which side is the leg adjacent to ∠X?
Answer: A
The adjacent leg is the leg that forms the angle: XY connects X to the right-angle vertex. Choice B, YZ, is opposite ∠X. Choice C, XZ, is the hypotenuse, which is also a side of ∠X but is never called the adjacent leg.
Question 10 of 20 · Multiple Choice
Why are sine, cosine and tangent in a right triangle defined only for its acute angles?
Answer: B
Each ratio needs a leg opposite the angle and a leg next to it. For the right angle, the side across from it is the hypotenuse, so the definitions do not apply. Choice C is false: a right triangle has two acute angles. Choice A is not the reason: the unit circle (HSF.TF.A.2) later gives obtuse angles a sine as well.
Question 11 of 20 · Multiple Choice
A student draws a right triangle with a 58° angle and a 12 cm hypotenuse. About how long should the leg opposite the 58° angle measure? (sin 58° ≈ 0.848)
Answer: D
opposite = 12 sin 58° ≈ 12(0.848) ≈ 10.2 cm. Choice A uses the cosine, which gives the adjacent leg. Choice B divides by the sine, and choice C uses the tangent.
Question 12 of 20 · Multiple Choice
For an acute angle A, tan A = 39/80. What is sin A?
Answer: C
Use a right triangle with legs 39 and 80. The hypotenuse is √(39² + 80²) = √(1521 + 6400) = √7921 = 89, so sin A = 39/89. Choice B is cos A. Choice A adds the legs instead of using the Pythagorean Theorem.
Question 13 of 20 · Multiple Choice
Which right triangle has the same sine for the angle opposite its shortest leg as a right triangle with sides 20, 48 and 52?
Answer: A
The triangle 20-48-52 is a scale copy of 5-12-13 (factor 4), and so is 25-60-65 (factor 5), so they are similar and the ratios match: 20/52 = 25/65 = 5/13. Choice B shares the leg 20 but is not similar: 20/29 is not 5/13. Choices C and D are scale copies of 3-4-5 and 8-15-17.
Question 14 of 20 · Multiple Choice
A student says, "sin A = 3/5 means the leg opposite A is 3 units long and the hypotenuse is 5 units long." What is the best response?
Answer: B
A trigonometric ratio fixes the shape, not the size: every similar right triangle has the same ratio. Choice A confuses a ratio with lengths. Choice C treats the ratio as a fraction of the angle, which it is not: 3/5 of 90° is 54°, but sin A = 0.6 gives A ≈ 36.9°.
Question 15 of 20 · Short Answer
Explain why tan 50° is the same number no matter which right triangle with a 50° angle you use to compute it.
Any two right triangles with a 50° angle have two pairs of congruent angles (the 50° angles and the right angles), so they are similar by AA. Corresponding sides are proportional: opposite₁/opposite₂ = adjacent₁/adjacent₂. Rearranging gives opposite₁/adjacent₁ = opposite₂/adjacent₂, so the tangent is the same in both. A calculator gives tan 50° ≈ 1.19.
Question 16 of 20 · Short Answer
A right triangle has legs 48 and 55. Find the hypotenuse, then write sin, cos and tan of the angle opposite the 48-unit leg.
Hypotenuse = √(48² + 55²) = √(2304 + 3025) = √5329 = 73. For the angle opposite 48: sin = 48/73, cos = 55/73, tan = 48/55. A common error is using 73 as the adjacent side in the tangent.
Question 17 of 20 · Short Answer
For an acute angle B, cos B = 60/109. Find sin B and tan B.
Sketch a right triangle with adjacent leg 60 and hypotenuse 109. The opposite leg is √(109² - 60²) = √(11881 - 3600) = √8281 = 91. So sin B = 91/109 and tan B = 91/60. Any triangle with the same angle B gives the same ratios, so using these particular side lengths is fine.
Question 18 of 20 · Short Answer
At the same time of day, a 1.5-meter post casts a 2-meter shadow and a tree casts an 18-meter shadow on level ground. Use a trigonometric ratio to find the height of the tree, and explain why the method works.
The sun's rays meet the level ground at the same angle for the post and the tree, and both stand vertically, so the two right triangles have a congruent acute angle and are similar. The tangent of the sun's angle is 1.5/2 = 0.75 in both, so height/18 = 0.75 and the tree is 13.5 meters tall.
Question 19 of 20 · Short Answer
Explain why sin A and cos A are always less than 1 for an acute angle A in a right triangle. Is the same true for tan A?
The hypotenuse is opposite the right angle, the largest angle, so it is the longest side. A leg divided by a longer side is less than 1, so sin A < 1 and cos A < 1. Tangent divides one leg by the other, so tan A can be greater than 1: it is greater than 1 whenever the opposite leg is longer than the adjacent leg, that is, when A is more than 45°.
Question 20 of 20 · Short Answer
A right triangle has legs 5 and 5. Find the hypotenuse and the sine, cosine and tangent of one acute angle. Explain why every isosceles right triangle gives the same three values.
Hypotenuse = √(25 + 25) = 5√2. For either acute angle: sin = cos = 5/(5√2) = √2/2 ≈ 0.707 and tan = 5/5 = 1. Every isosceles right triangle has angles 45°, 45° and 90°, so all of them are similar by AA and give the same ratios: these are sin 45°, cos 45° and tan 45°.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does HSG.SRT.C.6 mean?
HSG.SRT.C.6 means students understand that side ratios in a right triangle depend only on its angles, and that this fact is what defines sine, cosine and tangent. The reason is similarity: two right triangles with a congruent acute angle are similar by AA, so a ratio such as opposite leg over hypotenuse is the same in both. The ratio can then be named after the angle: sin A, cos A or tan A.
Is HSG.SRT.C.6 taught in Geometry or Algebra 2?
It is usually taught in Geometry, right after similarity (HSG.SRT.A.2 and HSG.SRT.A.3). Algebra 2 and Precalculus then extend sine and cosine to all angles with the unit circle (HSF.TF.A.2), but the right-triangle definitions come first.
Why do you need similar triangles to define sine?
Without similarity, "sin 35°" would not make sense as a single number, because you could draw many different right triangles with a 35° angle. Similarity shows that all of them give the same ratio, so the ratio really belongs to the angle. That is the whole point of the standard.
Do trig ratios change when the triangle gets bigger?
No. Scaling a right triangle multiplies every side by the same factor, and the factor cancels in any ratio of two sides. The angles do not change either. Only the side lengths change.
Is SOH-CAH-TOA enough to learn this standard?
No. SOH-CAH-TOA is a useful memory aid for which sides go in each ratio, but HSG.SRT.C.6 asks students to understand why the ratios are properties of the angle. Students should be able to explain the similarity argument, not only recite the mnemonic.
What mistakes do students make with opposite and adjacent sides?
Students often name the sides from the wrong angle, especially when the triangle is rotated or the angle is at the top. Another frequent error is calling the hypotenuse the adjacent side because it also touches the angle. Have students trace the angle's two sides with a finger: the one that is a leg is adjacent, and the leg across from the angle is opposite.
How is tangent related to slope?
Take a line that rises from left to right and draw a right triangle under any piece of it, with one horizontal leg and one vertical leg. Every such triangle has the same acute angle where it meets the line, so all of them are similar, and vertical leg divided by horizontal leg is the same number each time. That number is the slope of the line, and it is also the tangent of the angle the line makes with the horizontal.
How is this standard tested?
Typical tasks ask students to explain why two right triangles with the same acute angle have equal ratios, to write sin, cos and tan from given side lengths, or to pick the ratio that matches a labeled side. The digital SAT includes right-triangle trigonometry in its Geometry and Trigonometry domain, so this vocabulary is used there as well.
Do students need to memorize values like sin 30°?
Not for this standard. Students can find exact values from special triangles, as in the 30-60-90 practice in this lesson, and use a calculator in degree mode for other angles. Knowing the special values from memory is part of HSF.TF.A.3, a later standard.
What comes after HSG.SRT.C.6?
The next standards in the cluster use the definitions: HSG.SRT.C.7 explains why the sine of an angle equals the cosine of its complement, and HSG.SRT.C.8 uses trigonometric ratios and the Pythagorean Theorem to solve applied right-triangle problems. Later, the unit circle extends the same ratios beyond acute angles.
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Related Standards
5 standards
These standards connect to HSG.SRT.C.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSG.SRT.A.3Prerequisite
Use similarity transformations to establish the AA criterion for similar triangles
