Finding sides with trigonometry

About this lesson

Integrated Math II › Right triangle trigonometry

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Finding sides with trigonometry

Warm-up

Three quick questions from earlier lessons.

Discover

A kite string 30 m long makes a 40∘40^\circ angle with the ground. Which ratio gives the kite's height?

  1. sin⁡40∘=height30\sin 40^\circ = \dfrac{\text{height}}{30}
  2. cos⁡40∘=height30\cos 40^\circ = \dfrac{\text{height}}{30}

Key idea

Pick the ratio that links what you know with what you want.

Label the sides from the given angle, choose sin, cos or tan, write the equation, then solve: if the unknown is on top, multiply; if it is in the denominator, divide. Use degree mode.

Worked example

Kite: height =30sin⁡40∘= 30 \sin 40^\circ.

30×0.64330 \times 0.643
sin 40° ≈ 0.643.
height ≈19.28\approx 19.28 m
Round.

Your turn, guided

Your turn: hypotenuse 10, angle 30∘30^\circ. The side opposite.

55
10 × sin 30° = 10 × ½.

Practice ★

  1. A right triangle has hypotenuse 14 and an acute angle 43∘43^\circ. Find the side opposite that angle (to 2 decimal places).
  2. A right triangle has hypotenuse 19 and an acute angle 22∘22^\circ. Find the side opposite that angle (to 2 decimal places).
  1. A ramp rises at 55∘55^\circ over a horizontal distance of 15 m. How high does it rise? (2 decimal places)

Watch out!

Practice ★★

  1. The side opposite a 40∘40^\circ angle in a right triangle is 4. Find the hypotenuse (2 decimal places).
  2. The side opposite a 40∘40^\circ angle in a right triangle is 9. Find the hypotenuse (2 decimal places).
  1. Boss problem. A 8 m ladder leans against a wall at 55∘55^\circ to the ground. Give answers to 2 decimal places.

Exit ticket

Three questions to check you have got it.

  1. A right triangle has hypotenuse 14 and an acute angle 43∘43^\circ. Find the side opposite that angle (to 2 decimal places).
  2. A right triangle has hypotenuse 19 and an acute angle 22∘22^\circ. Find the side opposite that angle (to 2 decimal places).
  3. A right triangle has hypotenuse 18 and an acute angle 58∘58^\circ. Find the side opposite that angle (to 2 decimal places).

Summary card

Finding sides

Label → choose the ratio → solve (multiply or divide).

Watch out: the hypotenuse is the longest side; check your answer.

End of the lesson

Next lesson: Finding angles with inverse trigonometry Back to the chapter

Answers: Finding sides with trigonometry

A

  1. 14sin⁡(43π180)14 \sin{\left(\frac{43 \pi}{180} \right)}
  2. 19sin⁡(11π90)19 \sin{\left(\frac{11 \pi}{90} \right)}

B

  1. 15tan⁡(11π36)15 \tan{\left(\frac{11 \pi}{36} \right)}

C

  1. 4sin⁡(2π9)\frac{4}{\sin{\left(\frac{2 \pi}{9} \right)}}
  2. 9sin⁡(2π9)\frac{9}{\sin{\left(\frac{2 \pi}{9} \right)}}

D

  1. 3535

E

  1. 14sin⁡(43π180)14 \sin{\left(\frac{43 \pi}{180} \right)}
  2. 19sin⁡(11π90)19 \sin{\left(\frac{11 \pi}{90} \right)}
  3. 18sin⁡(29π90)18 \sin{\left(\frac{29 \pi}{90} \right)}

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