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Trigonometry: Right Triangles, SOH CAH TOA and the Unit Circle

Trig ratios, the Pythagorean theorem, special angles, solving right triangles, radians, the unit circle and the law of cosines. Geometry and precalculus level.

by Kill h.17 questionsCreated Nov 8, 2026, 08:04 PM

Geometry & Trigonometry· Trigonometric Functions#trigonometry#soh-cah-toa#right-triangles#unit-circle

17 questions

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1
In the right triangle shown, what is ?
- A
- C
- D
SOH CAH TOA: Sine = Opposite / Hypotenuse. Adjacent / hypotenuse is , and opposite / adjacent is .
2
In the right triangle shown, what is ?
- A
- B
- D
TOA: Tangent = Opposite / Adjacent. It also equals . Adjacent / opposite is .
3
In the triangle shown, the opposite side is 3 and the adjacent side is 4. How long is the hypotenuse?
- A
  7
- B
  12
- C
- 5
Pythagorean theorem: , so c = 5. Adding the legs (7) is the usual slip: you add the squares, not the sides.
4
With opposite = 3, adjacent = 4 and hypotenuse = 5, what is ?
- B
- C
- D
Cosine = adjacent / hypotenuse = 4/5. is and is .
5
A right triangle has legs of 5 and 12. What is the hypotenuse?
- A
  17
- 13
- C
  11
- D
. 5-12-13 is a Pythagorean triple worth memorizing, along with 3-4-5 and 8-15-17. comes from subtracting the squares.
6
What is the exact value of ?
- A
- B
- C
In a 30-60-90 triangle with sides , the side adjacent to 30° is , so . is (or ).
7
A 10 m ladder leans against a wall, making a 60° angle with the ground. How high up the wall does it reach?
- About 8.66 m
- B
  5.0 m
- C
  About 17.3 m
- D
  About 11.5 m
The height is opposite the 60° angle and the ladder is the hypotenuse: . 5.0 m uses cosine, which gives the distance from the wall.
8
A building is 50 m tall. From a point 50 m from its base on level ground, what is the angle of elevation to the top?
- A
  30°
- B
  60°
- 45°
- D
  90°
, so . Equal legs always mean a 45-45-90 triangle.
9
A right triangle has a hypotenuse of 20 and an angle of 35°. How long is the side opposite the 35° angle?
- About 11.5
- B
  About 16.4
- C
  About 14.0
- D
  About 34.9
. 16.4 uses cosine (that is the adjacent side), and 14.0 uses tangent.
10
Convert 135° to radians.
- B
- C
- D
Multiply by : . is 270°.
11
What is ?
- A
- C
- D
210° is in quadrant III with a reference angle of 30°. Sine is negative in quadrant III, so . is .
12
If and is in quadrant II, what is ?
- A
- B
- D
, so and . Cosine (the x-coordinate) is negative in quadrant II, so .
13
A triangle has sides a = 7 and b = 8 with an included angle C = 60°. How long is side c?
- A
- B
- C
  15
Law of cosines: , so . treats the triangle as if it had a right angle.
14
Two sides of a triangle are 8 and 10, and the angle between them is 30°. What is the area?
- A
  40
- 20
- C
  80
- D
. 40 forgets the sine factor, and uses cosine.
15
A circle has a radius of 6. What is the length of the arc cut off by a central angle of radians?
- A
- B
- C
Arc length . The formula only works with the angle in radians.
16
A 45-45-90 triangle has legs of length 5. How long is the hypotenuse?
- B
- C
- D
The sides of a 45-45-90 triangle are in the ratio , so the hypotenuse is . Check: . belongs to a 30-60-90 triangle.
17
What is the period of ?
- A
- B
- C
For , the period is . Multiplying x by 2 squeezes the graph horizontally, so it repeats twice as often. would be the period of .