Lesson 9

Using Trigonometric Ratios to Find Angles

  • Let’s work backwards to find angles in right triangles.

Problem 1

Technology required. Ramps in a parking garage need to be both steep and safe. The maximum safe incline for a ramp is 8.5 degrees. Is this ramp safe? If not, provide dimensions that would make the ramp safe.

Car driving down right triangle ramp. Legs of right triangle are 15 and 95.

Problem 2

Technology required. \(ABCD\) is a rectangle. Find the length of \(AC\) and the measures of \(\alpha\) and \(\theta\)

Rectangle A B C D with diagonal A C. AB is 3 units, B C is 12 units. Angle B A C is labeled alpha and A C B is labeled theta.

Problem 3

Technology required. Find the missing measurements. 

Right triangle abc. Side AB = 21 units, side CA= 35 units. Hypotenuse unknown 

Problem 4

Select all the true equations:

Right triangle A B C. A C is x units, B C is y units, A B is 15 units. Angle B A C is 27 degrees, angle A B C is 63 degrees, and angle A C B is 90 degrees.

\(\sin(27) =\frac{x}{15}\)


\(\cos(63) =\frac{y}{15}\)


\(\tan(27) = \frac{y}{x}\)


\(\sin(63) = \frac{x}{15}\)


\(\tan(63) = \frac{y}{x}\)

(From Unit 4, Lesson 8.)

Problem 5

What value of \(\theta\) makes this equation true? \(\sin(30)=\cos(\theta)\)









(From Unit 4, Lesson 8.)

Problem 6

A rope with a length of 3.5 meters is tied from a stake in the ground to the top of a tent. It forms a 17 degree angle with the ground. How tall is the tent?


\(3.5 \tan(17)\)


\(3.5 \cos(17)\)


\(3.5 \sin(17)\)



(From Unit 4, Lesson 7.)

Problem 7

Technology required. What is the value of \(x\)

Triangle D E F. Angle D is 40 degrees. Angle E is 50 degrees. Side D E is x. Side F D is 3.
(From Unit 4, Lesson 6.)

Problem 8

Find the missing side in each triangle using any method. Check your answers using a different method.

2 right triangles. On left, base = 3 units, height = x, hypotenuse = 5 units. On right, base = 9 units, height = 12 units, hypotenuse = y.
(From Unit 4, Lesson 1.)

Problem 9

The triangles are congruent. Write a sequence of rigid motions that takes triangle \(XYZ\) onto triangle \(BCA\).

Triangle ABC and ZXY.

(From Unit 2, Lesson 3.)