Question

Answer any of the following if you know one but not the other!!

1. If the terminal side of angle θ, in standard position, passes through the point (4, -7), what is the numerical value of sinθ?

2. Given tanθ = 3/2 and angle θ is in quadrant III, find the exact value of sinθ in simplest radical form using a rational denominator.

3. Why is it true that sin(θ) = cos (90 - θ) and cos(θ) = sin (90 - θ)?

Answer 1

Explanation:

1.

the length of the terminal side from (0, 0) to (4, -7) is the radius of the circle, while

x = 4 = cos(theta)×radius

y = -7 = sin(theta)×radius

so,

sin(theta) = -7/radius

radius² = (4 - 0)² + (-7 - 0)² = 4² + 7² = 16 + 49 = 65

radius = sqrt(65)

sin(theta) = -7/ sqrt(65) = -0.868243142...

FYI

theta = -60.2551187...°

or 299.7448813...°

2.

tan(theta) = 3/2

so,

sin(theta)/cos(theta) = 3/2

2×sin(theta) = 3×cos(theta)

square both sides

4×sin²(theta) = 9×cos²(theta)

completing the square

4×sin²(theta) + 9×sin²(theta) = 9×cos²(theta) + 9×sin²(theta)

13×sin²(theta) = 9×(cos²(theta) + sin²(theta)) = 9

sin²(theta) = 9/13

sin(theta) = 3/sqrt(13)

to make it a rational denominator we multiply by

sqrt(13)/ sqrt(13)

and we get

sin(theta) = 3×sqrt(13)/13

3.

because the sum of all angles in a triangle is 180°.

one angle is theta.

then we have the 90° angle at the origin (or center of the circle).

and that leaves 180 - 90 - theta = 90 - theta for the third angle.

this triangle can be flipped with putting 90 - theta in place of theta.

the flipped triangle is equivalent to the original triangle (all angles and side lengths are identical between the 2 triangles).

the baseline (radius) is the same length in both triangles.

in that flipped triangle the side formerly sin(theta) is now cos(90-theta) and cos(theta) is now sin(90-theta).