Lesson 9

Using Trigonometric Ratios to Find Angles

The practice problem answers are available at one of our IM Certified Partners

Problem 1

Technology required. Ramps in a parking garage need to be steep. 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 Geometry, Unit 4, Lesson 8.)

Problem 5

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









(From Geometry, 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 Geometry, 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 Geometry, 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 Geometry, 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 Geometry, Unit 2, Lesson 3.)