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Math 10C

Math 10C · Worksheets

Right-triangle trigonometry

Ten questions of mixed difficulty, covering Trigonometry. Print it, or work through it on screen — the answer key starts on its own page.

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Right-triangle trigonometry

Math 10C · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. A ladder 6 m long leans against a wall and reaches 5 m up it. The angle it makes with the ground is about

    1. a) 34°
    2. b) 40°
    3. c) 50°
    4. d) 56°

  2. A right triangle has an angle of 35° and the side adjacent to it is 8 cm. The opposite side is about

    1. a) 5.6 cm
    2. b) 6.6 cm
    3. c) 4.6 cm
    4. d) 11.4 cm

  3. In a right triangle, cos θ = 0.6. The angle θ is about

    1. a) 31°
    2. b) 0.6°
    3. c) 53°
    4. d) 37°

  4. In a right triangle the side opposite θ is 5 and the hypotenuse is 13. Then sin θ is

    1. a) 5/12
    2. b) 12/13
    3. c) 5/13
    4. d) 13/5

  5. As an acute angle θ increases towards 90°, its tangent

    1. a) approaches 0
    2. b) stays the same
    3. c) approaches 1
    4. d) increases without limit

  6. A right triangle has legs 9 and 12. The smallest angle is about

    1. a) 41°
    2. b) 45°
    3. c) 53°
    4. d) 37°

  7. In a right triangle, the side <i>adjacent</i> to an angle is

    1. a) always the horizontal side
    2. b) the side across from the angle
    3. c) the shorter side next to that angle, not the hypotenuse
    4. d) the longest side

  8. In a right triangle, the hypotenuse is

    1. a) the side next to the angle you are using
    2. b) always the vertical side
    3. c) the side opposite the angle you are using
    4. d) the side opposite the right angle, and always the longest

  9. sin θ is defined as

    1. a) adjacent ÷ hypotenuse
    2. b) opposite ÷ hypotenuse
    3. c) hypotenuse ÷ opposite
    4. d) opposite ÷ adjacent

  10. tan θ is defined as

    1. a) adjacent ÷ opposite
    2. b) adjacent ÷ hypotenuse
    3. c) opposite ÷ hypotenuse
    4. d) opposite ÷ adjacent

Answer key · Right-triangle trigonometry

Math 10C · maddyhelps.com

  1. d) 56° — The height is opposite the angle and the ladder is the hypotenuse, so sin θ = 5/6 and θ = sin⁻¹(0.833) ≈ 56°. Drawing the triangle and labelling the sides before choosing the ratio is the whole method.
  2. a) 5.6 cm — Opposite and adjacent means tangent: tan 35° = x/8, so x = 8 tan 35° ≈ 5.6 cm. The opposite side should be shorter than the adjacent one because 35° is less than 45°, which is a useful check.
  3. c) 53° — θ = cos⁻¹(0.6) ≈ 53.1°. Make sure the calculator is in degree mode — in radians the same keystrokes give 0.93, which looks like a plausible answer and is not.
  4. c) 5/13 — Sine is opposite over hypotenuse, so 5/13. The third side is 12 by Pythagoras, which would give cos θ = 12/13 and tan θ = 5/12 — but the question asked only for sine.
  5. d) increases without limit — tan θ = opposite/adjacent. As the angle opens up, the opposite side grows while the adjacent side shrinks, so the ratio grows without bound. At exactly 90° the adjacent side is zero and the tangent is undefined.
  6. d) 37° — The smallest angle is opposite the shortest side, so tan θ = 9/12 and θ ≈ 36.9°. Checking which angle you have found against which side is opposite it catches the common mistake of reporting 53°.
  7. c) the shorter side next to that angle, not the hypotenuse — Both the hypotenuse and the adjacent side touch the angle, so adjacent is defined as the one that is not the hypotenuse. Labelling all three sides before choosing a ratio prevents most errors in this unit.
  8. d) the side opposite the right angle, and always the longest — The hypotenuse never changes as you switch which acute angle you are working from — it is fixed by the right angle. The opposite and adjacent sides do swap over, which is why labelling has to be done for one angle at a time.
  9. b) opposite ÷ hypotenuse — SOH: Sine is Opposite over Hypotenuse. CAH and TOA give the other two. The ratio depends only on the angle, which is why one table of values works for every right triangle.
  10. d) opposite ÷ adjacent — TOA. Tangent is the only one of the three that does not involve the hypotenuse, which makes it the right choice whenever the hypotenuse is neither known nor wanted.