Right triangle trigonometry: flashcards, quiz and summary

Flashcards

1. 1What does the Pythagorean theorem say, and when can you use it?

   For a right triangle with legs a and b and hypotenuse c, a² + b² = c². You can use it only in right triangles, and c must be the hypotenuse.

2. 2How do you decide which leg is opposite and which is adjacent?

   Pick the acute angle you are working with. The opposite leg is across from that angle, and the adjacent leg touches that angle without being the hypotenuse. The labels change if you switch to the other acute angle.

3. 3How are sine, cosine, and tangent defined in terms of the sides?

   sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. SOH CAH TOA is the memory aid.

4. 4How do you find a missing side when you know one acute angle and one side?

   Label the sides from the known angle, choose the ratio that contains the known side and the unknown side, write the equation, and solve for the unknown. For example, if the hypotenuse is 10 and the angle is 40°, the opposite side is 10 × sin(40°), about 6.43.

5. 5How do you find an acute angle when you know two sides of a right triangle?

   Choose the ratio that uses those two sides, then apply the inverse function. For example, if the opposite side is 5 and the adjacent side is 12, then θ = tan⁻¹(5/12), about 22.6°. The other acute angle is 90° minus that value.

6. 6What are the side ratios in 45-45-90 and 30-60-90 triangles?

   In a 45-45-90 triangle, the legs are equal and the sides are x : x : x√2. In a 30-60-90 triangle, the sides are x : x√3 : 2x, with the short leg across from 30°, the long leg across from 60°, and the hypotenuse equal to 2x.

7. 7How are the angle of elevation and the angle of depression related?

   The angle of elevation measures up from the horizontal to an object above, and the angle of depression measures down from the horizontal to an object below. The angle of elevation from A to B equals the angle of depression from B to A.

8. 8How are cosecant, secant, and cotangent defined, and how do they relate to the basic ratios?

   csc θ = 1/sin θ = hypotenuse/opposite, sec θ = 1/cos θ = hypotenuse/adjacent, and cot θ = 1/tan θ = adjacent/opposite. Each is the reciprocal of one of the three main ratios.

9. 9Why does sin⁻¹(x) not mean the same thing as 1/sin(x)?

   sin⁻¹(x) is the inverse sine, which returns an angle whose sine is x. 1/sin(x) is the reciprocal, which equals csc(x). The notation looks similar but the operations are different.

Practice quiz

Pick an answer to check it.

1. 1A right triangle has legs of 9 and 12. What is the hypotenuse?

   1. 13
   2. 21
   3. 225
   4. 15
   Show the answer

   D. 15 9² + 12² = 81 + 144 = 225, and the square root of 225 is 15. Taking the square root is the step students most often skip.

2. 2In a right triangle, one acute angle is 40° and the hypotenuse is 10. What is the length of the side opposite the 40° angle, to two decimal places?

   1. 7.66
   2. 3.21
   3. 6.43
   4. 8.39
   Show the answer

   C. 6.43 The opposite side and the hypotenuse are linked by sine, so the side is 10 × sin(40°) ≈ 10 × 0.6428 ≈ 6.43. Cosine would give the adjacent side, and tangent would need the adjacent side, not the hypotenuse.

3. 3In a 3-4-5 triangle, for the angle across from the side of length 4, what is tan θ?

   1. 4/3
   2. 3/5
   3. 3/4
   4. 4/5
   Show the answer

   A. 4/3 Tangent is opposite over adjacent. The opposite side to this angle is 4, and the adjacent side is 3, so tan θ = 4/3. Students often flip the fraction or use the hypotenuse by mistake.

4. 4A 30-60-90 triangle has a hypotenuse of 12. What is the length of the long leg?

   1. 12√3
   2. 6√3
   3. 4√3
   4. 6
   Show the answer

   B. 6√3 The sides are x : x√3 : 2x. With 2x = 12, x = 6, so the long leg is 6√3, about 10.39. The short leg is 6, and the long leg is the one with √3.

5. 5You know the opposite and adjacent sides of a right triangle relative to an acute angle, and you want the angle. Which function should you use?

   1. tan⁻¹
   2. sin⁻¹
   3. cos⁻¹
   4. csc⁻¹
   Show the answer

   A. tan⁻¹ Tangent uses opposite and adjacent, so its inverse, tan⁻¹, finds the angle. sin⁻¹ needs the opposite and hypotenuse, and cos⁻¹ needs the adjacent and hypotenuse.

6. 6A 45-45-90 triangle has legs of 5. What is the hypotenuse?

   1. 10
   2. 5√3
   3. 2.5√2
   4. 5√2
   Show the answer

   D. 5√2 In a 45-45-90 triangle, the sides are x : x : x√2. With x = 5, the hypotenuse is 5√2, about 7.07. Doubling the leg gives 10, which is not how the ratio works.

7. 7What is sin⁻¹(0.5)?

   1. 60°
   2. 30°
   3. 1/sin(0.5)
   4. 2
   Show the answer

   B. 30° sin⁻¹(0.5) is the angle whose sine is 0.5, which is 30°. The value 60° is the angle whose cosine is 0.5, and 1/sin(0.5) is a reciprocal, not an inverse.

Test traps

What teachers like to test on this, and the mistakes students make.

• Mixing up opposite and adjacent when the problem uses the other acute angle. The labels depend on the angle, so relabel after you switch angles.
• Using the Pythagorean theorem with a leg in the c position. Only the hypotenuse goes on the right side of a² + b² = c².
• Confusing sin⁻¹(x) with 1/sin(x). The inverse returns an angle, while the reciprocal is csc(x).
• Forgetting to set the calculator to degree mode, which gives wrong angle and side answers.
• In a 30-60-90 triangle, putting √3 on the short leg instead of the long leg, or doubling the short leg incorrectly when finding the hypotenuse.
• Forgetting the last step of a Pythagorean problem: taking the square root, so the answer is c² instead of c.

Summary

• In a right triangle, the hypotenuse is the longest side and sits across from the right angle, while the two legs form the right angle.
• The Pythagorean theorem a² + b² = c² finds a missing side when you know the other two, and c must always be the hypotenuse.
• The ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent, which SOH CAH TOA helps you remember.
• Opposite and adjacent depend on which acute angle you are using, but the hypotenuse never changes.
• To find a missing side, pick the ratio that contains the known side and the unknown side, then solve. To find an angle, use the inverse functions sin⁻¹, cos⁻¹, or tan⁻¹ in degree mode.
• Special triangles have fixed side ratios: 45-45-90 has x : x : x√2, and 30-60-90 has x : x√3 : 2x.
• Angles of elevation and depression are measured from the horizontal, and they are equal when viewed between the same two points.

Key terms

Hypotenuse

The side across from the right angle. It is always the longest side of a right triangle.

Legs

The two sides that form the right angle of a right triangle.

Pythagorean triple

A set of whole numbers a, b, c that satisfies a² + b² = c², such as 3-4-5 or 5-12-13.

Opposite side

The leg across from the acute angle θ you are working with.

Adjacent side

The leg next to the acute angle θ that is not the hypotenuse.

Inverse trig function

sin⁻¹, cos⁻¹, and tan⁻¹ return an angle when you know a ratio of sides. sin⁻¹(x) is not the same as 1/sin(x).

Angle of elevation

The angle measured up from the horizontal to an object above the horizontal line of sight.

Angle of depression

The angle measured down from the horizontal to an object below the horizontal line of sight.

How to study it

The notes this pack came from

Questions