Answer to Question #126494 in Trigonometry for Deborah

Question

1. Evaluate the following:
a. Cos 118°
b. Sin 135°
c. Tan 252°
d. Sin (-60°)
e. Tan (-148°)

2. Determine the following ratios:

a. Sin 850°
b. Cos 920°
c. Tan 1040°

Answer 1

Question 1

Final answers that cannot be given in exact form are provided to 3 significant figures.

a) Cos118°

118° is in the second quadrant and hence the ratio Cos118° is negative.

Thus,

Cos118° = -Cos(180° - 118°)

= -Cos62°

= -0.469471563

= -0.469

b) Sin135°

The angle 135° is in the second quadrant and hence the value of the ratio Sin135° is positive.

Thus,

Sin135° = Sin(180° - 135°)

= Sin45°

= $√2/2$

c) Tan252°

The angle 252° is in the third quadrant and hence the ratio Tan252° is positive.

Thus,

Tan252° = Tan(252° - 180°)

= Tan72°

= 3.077683537

= 3.08

d) Sin(-60°)

Negative angles imply that an object was rotated in a clockwise direction from the reference point.

The angle -60° is therefore equal to 360° - 60° = 300°.

The angle 300° is in the third quadrant and hence the ratio Sin(-60°) is negative.

Thus,

Sin(-60°) = -Sin60°

= -$√3/2$

e) Tan(-148°)

-148° implies that the object was rotated 148° in a clockwise direction. As a result -148° = 180° - 148°

= 32°

32° is in the first quadrant and hence the ratio Tan(-148°) is positive.

Thus,

Tan(-148°) = Tan32°

= 0.624869352

= 0.625

Question 2

The angles provided are more than 360°. Such angles imply that an object was rotated more than one complete revolution about a fixed point. To determine the basic trigonometric ratios of such angles, a relevant multiple of 360° is subtracted from the angle in question until a positive angle that is less than 360° is obtained. After that, appropriate trigonometric concepts are applied to determine the trigonometric ratio in question.

a) Sin850°

360° divides into 850° twice, therefore there are two full revolutions.

850° = 850° - (360° × 2)

= 850° - 720°

= 130°

Therefore, Sin850° = Sin130°

130° is in the second quadrant and hence the ratio Sin130° is positive.

Thus,

Sin850° = Sin130°

= Sin(180° - 130°)

= Sin50°

= 0.766044443

= 0.766

b) Cos920°

360° divides into 920° twice and hence there are two complete cycles involved. As a result, 920° = 920° -(360°×2)

= 920° - 720°

= 200°

200° is in the third quadrant and hence the ratio Cos920° is negative.

Thus,

Cos920° = Cos(920° - 720°)

= Cos200°

= -Cos(200° - 180°)

= -Cos20°

= -0.939692621

= -0.940

c) Tan1040°

There are two complete cycles in 1040° and as a result 1040° = 1040° - 720°

= 320°

320° is in the forth quadrant and hence the ratio Tan1040° is negative.

Thus,

Tan1040° = Tan(1040° - 720°)

= Tan320°

= -Tan(360° - 320°)

= -Tan40°

= -0.839099631

= -0.839