LABELLING TRIANGLES OPPOSITE ADJACENT HYPOTENUSE

Calculate the angle θ in each of the following triangles. In each case give your answer correct to 1 decimal place.

Problem 1 :

Solution :

Opposite side = 14 cm and Hypotenuse = 20 m

sin θ = Opposite side / hypotenuse

sin θ = 14/20

sin θ = 0.7

θ = sin^-1(0.7)

= 44.4

Approximately 44 degree.

Problem 2 :

Solution :

Opposite side = 7 cm and Adjacent side = 8 m

tan θ = Opposite side / Adjacent side

tan θ = 7/8

tan θ = 0.875

θ = tan^-1(0.875)

= 41.18

Approximately 41 degree.

Problem 3 :

The diagram shows the cross section of the shed. Calculate the angle θ between the roof and the horizontal. Give your answer to the nearest degree.

Solution :

At the bottom, we have rectangle. At the top, we have triangle.

Height of the triangle = 2.6 - 2.1

= 0.5

Base of the triangle = 3.4 m

In the right triangle above, Opposite side = 2.1 m and adjacent side = 3.4 m

tan θ = Opposite side / Adjacent side

tan θ = 2.1/3.4

= 0.617

θ = tan^-1(0.617)

= 31.67

Approximately 32 degree.

Problem 4 :

The ladder of length 6 m leans against a wall. The foot of the ladder is at a distance of 3 m from the base of the wall. Calculate the angle between the ladder and the ground.

Solution :

cos x = Adjacent side / hypotenuse

cos x = 3/6

x = cos^-1(0.5)

x = 30

So, the required angle between the ladder and the ground is 30 degree.

Problem 5 :

You are measuring the height of a lamppost. You stand 40 inches from the base of the lamppost. You measure the angle of elevation from the ground to the top of the lamppost to be 70°. Find the height h of the lamppost to the nearest inch.

Solution :

Opposite side = h inches

Adjacent side = 40 inches

tan θ = Opposite side / Adjacent side

tan 70 = h/40

2.74 = h/40

h = 2.74(40)

= 109.6

Problem 6 :

Describe the error in the statement of the tangent ratio. Correct the error if possible. Otherwise, write not possible.

Solution :

When D is the angle measure,

Opposite side = EF = 35

Adjacent side = DE = 12

Hypotenuse = 37

tan D = 35/12

tan θ is the ratio between opposite side by adjacent side. But the given ratio shows the ratio between opposite side by hypotenuse. That is the error.

Problem 7 :

Describe the error in the statement of the tangent ratio. Correct the error if possible. Otherwise, write not possible.

Solution :

Sum of interior angle measure = 180

∠A + ∠B + ∠C = 180

30 + ∠B + 55 = 180

125 + ∠B = 180

∠B = 180 - 125

∠B = 75

From this, it is clear that the given triangle is not a right triangle. So, we cannot use the tangent ratio. So, it is not possible.

Problem 8 :

A surveyor is standing 118 feet from the base of the Washington Monument. The surveyor measures the angle of elevation from the ground to the top of the monument to be 78°. Find the height h of the Washington Monument to the nearest foot.

Solution :

tan θ = Opposite side / Adjacent side

tan 78 = h/118

4.70 = h/118

h = 4.70(118)

= 554.6

Approximately 555 feet

So, the height of Washington Monument is 555 feet.

Problem 9 :

Scientists can measure the depths of craters on the moon by looking at photos of shadows. The length of the shadow cast by the edge of a crater is 500 meters. The angle of elevation of the rays of the Sun is 55°. Estimate the depth d of the crater.

Solution :

tan θ = Opposite side / Adjacent side

tan 55 = d/500

1.42 = d/500

d = 1.42(500)

d = 710

So, depth of the crater is 710 m.

Problem 10 :

Your family room has a sliding-glass door. You want to buy an awning for the door that will be just long enough to keep the Sun out when it is at its highest point in the sky. The angle of elevation of the rays of the Sun at this point is 70°, and the height of the door is 8 feet. Your sister claims you can determine how far the overhang should extend by multiplying 8 by tan 70°. Is your sister correct? Explain.

Solution :

Let x be the required length of overhang

tan 70 = 8/x

x = 8/tan 70

But your friend is deciding to multiply 8 and tan 70. So, she is not correct.

Problem 11 :

You are skiing on a mountain with an altitude of 1200 feet. The angle of depression is 21°. Find the distance x you ski down the mountain to the nearest foot.

Solution :

sin 21 = 1200/x

x = 1200/sin 21

x = 1200/0.358

x = 3351.9

Approximately 3352 ft