FIND THE MISSING SIDES OF THE GIVEN RIGHT TRIANGLES WITH TRIGONOMETRY

Problem 1 :

Find the length of x.

a) 33. 8       b) 48.3       c) 84.3      d) 72.0

Solution :

sin θ = opposite/hypotenuse

sin 55˚ = x/59

0.819 = x/59

x = 59 × 0.819

x = 48.3

So, option (b) is correct.

Problem 2 :

The line of sight from a small boat to the light at the top of a 35-foot lighthouse on a cliff 25 feet above the water makes a 25˚ angle with the water. To the nearest foot, how far is the boat from the cliff?

a) 141 feet      b) 128 feet      c) 27 feet       d) 75 feet

Solution :

According to the image information, this is a right angle triangle.

35 ft + 25 ft = 60 ft

tan θ = opposite/adjacent

tan 25˚ = 60/x

0.466 = 60/x

x = 128.75 feet

So, option (b) is correct.

Problem 3 :

Find the measure of x in the right triangle.

a) 22.4˚       b) 67.6˚     c) 20.9˚    d) 69.1˚

Solution :

cos θ = adjacent/ hypotenuse

cos x = 8/21

x = cos^-1 (8/21)

x = 67.6˚

So, option (b) is correct.

Problem 4 :

Find the angle measure to the nearest tenth of a degree:

θ = tan^-1 7.9321

a) 7.2˚      b) 82.8˚        c) 1.4˚       d) 0.1˚

Solution :

θ = tan^-1 (7.9321)

θ = 82.8˚

So, option (b) is correct.

Problem 5 :

Find the angle measure to the nearest tenth of a degree:

θ = sin^-1 0.2026

a) 0.2˚   b) 11.7˚   c) 78.3˚   d) 1.4˚

Solution :

θ = sin^-1 (0.2026)

θ = 11.7˚

So, option (b) is correct.

Problem 6 :

Find the missing side of the right triangle.

a) 3.3˚      b) 3.1˚      c) 24.7˚     d) 8.5˚

Solution :

tan θ = opposite/adjacent

tan 20˚ = 9/x

0.36 = 9/x

x = 9/0.363

x = 24.7

So, option (c) is correct.

Problem 7 :

Which of the following is NOT true for all values of θ?

a) cos θ = cos (90 + θ)        b)  (cos θ)² + (sin θ)² = 1

c) tan θ = sin θ/cos θ          d) sin θ = cos (90 - θ)

Solution :

cos θ = cos (90 + θ)

cos (90 + θ) = -sin θ

So, option (a) is correct.

Problem 8 :

Fill in the blank :

sin 4˚/cos 4˚ = tan ____

Solution :

sin θ/cos θ = tan θ

sin 4˚/cos 4˚ = tan 4˚

Problem 9 :

a) 30   b) 60   c) 70   d) 85

Solution :

tan θ = opposite/adjacent

tan x = 19/11

x = tan^-1(19/11)

x = 59.93˚

x =60˚

So, option (b) is correct.

Problem 10 :

A large totem pole in the state of Washington is 100 feet tall. At a particular time of day, the totem pole casts a 249 foot long shadow. Find the measure of ∠A to the nearest degree.

a) 68˚   b) 45˚   c) 35˚   d) 22˚

Solution :

tan θ = opposite/adjacent

tan A = 100/249

 A = tan^-1(100/249)

A = 21.88˚

So, option (d) is correct.

Problem 11 :

Write the ratios for sin A and cos A.

a. sin A = 3/5, cos A = 4/5         b. sin A = 4/5, cos A = 3/5

c. sin A = 3/4, cos A = 4/5        d. sin A = 3/5, cos A = 4/3

Solution :

Opposite side = 4

Adjacent side = 3

Hypotenuse = 5

sin θ = opposite/hypotenuse

Sin A = 4/5

cos θ = adjacent/ hypotenuse

cos A = 3/5

So, option (b) is correct.

Problem 12 :          

What is the value of sin 43˚ to the nearest ten-thousandth?

a) 0.9325           b) 0.7314         c) 1.4663           d) 0.682

Solution :

Sin 43˚ = 0.682

So, option (d) is correct.

Problem 13 :

What is cos B for the triangle shown?

a)  8/17   b)  15/17   c)  8/15   d)  17/8

Solution :      

Opposite side = 15

Adjacent side = 8

Hypotenuse = 17

cos θ = adjacent/ hypotenuse

Cos B = 8/17

So, option (a) is correct.