The point (15, 20) is on the terminal side of an angle drawn in standard position. Find the exact value of the cosine of the angle.

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Relevance

It is a right angle triangle with base = adjacent side = 15 and height = opposite side = 20.

cos (angle) = adjacent / hypotenuse

So, first you have to use the Pythagorean theorem to find the hypotenuse (i.e. hypotenuse^2 = adjacent^2 + opposite^2)

Once you find hypotenuse, substitute it into the cosine equation listed above to find the angle using a calculator.

• Trigonometry

Angles & Circular Functions: Circular Functions

Defining Sine and Cosine

The trig functions are sine, cosine, tangent, cotangent, secant, & cosecant. Trig functions are evaluated at angles. For example, the sine of a 30° angle equals 0.5. The value of a trig function at an angle is based on what the sides of a triangle would be if that were the angle in the triangle. To explore these functions, we will start with sine and cosine.

To define sine and cosine, we will place the angle,θ, in standard position on an x-y coordinate plane. Pick any point on the terminal side and label it (x, y).

For any angle,θ, in standard position with a point (x, y) on its terminal side, the distance from the origin to the point (x, y) being r=x2+y2, the sine and cosine are

cosθ=xr and sinθ=yr.

Example: Find the sine and cosine of an angle,θ, in standard position if the point (3, 4) lies on its terminal side. Sketch the angle.

642-2-4-10-55θr(3, 0) (3, 4) r=x2+y2r=9+16r=5 Calculate r: For the angle in this position, x = 3, y = 4 and r = 5. cosθ=xr=35sinθ=yr=45

Example 2: Find sinθ if cosθ=817 and the terminal side of the angle θ is in the first quadrant. Sketch the angle:

2015105-5-10-15-10102030

θy817

The Unit Circle

Since there are infinite points, (x, y), on the terminal side of the angle,θ, we may as well choose the easiest point to use in our calculations. We will use the points that have r = 1. Every angle has a point at this radius. These points form the Unit Circle (the radius of the circle is one unit). With r = 1, the definitions of sine and cosine change.

If the terminal side of an angle θ in standard position intersects the unit circle at a point (x , y ), then

cosθ=x and sinθ=y.

Below is drawn a circle with radius 1. An angle,θ, is shown on the plane. Since r = 1, the point where the terminal side intersects the circle, (x, y) can be translated into cosθ, sinθ().

1-1-2 2

θ

While it may be a challenge to write the ordered pairs for most of the points on this circle, there are four points that we can label easily. 21-1-2(0,-1)(-1,0)(0,1)(1,0)The Unit Circle

Use the diagram above to calculate the following.

1. Find the sine of 90° (abbreviated sin 90°). Imagine a 90° angle in standard position. Where does the terminal side land? What is the ordered pair at the end of the terminal side? The sine of 90° is the y-value of that ordered pair!

sin 90° = 1

2. Find cos π: Imagine a π (radians) angle in standard position. Where does the terminal side land? What is the ordered pair at the end of the terminal side? The cosine of π is the x-value of that ordered pair!

cos π = - 1

3. You try these:

a. cos 270° = _________

b. cos π2 = _________

c. sin 4π = _________

d. sin 9π2 = _________

The Six Trigonometric Functions

The trig functions are sine, cosine, tangent, cotangent, secant, & cosecant. To define all six trig functions, we will place the angle,θ, in standard position on an x-y coordinate plane. Pick any point on the terminal side and label it (x, y).

For any angle,θ, in standard position with a point (x, y) on its terminal side, the distance from the origin to the point (x, y) being r=x2+y2, the trig functions are defined as the following:

sinθ=yr cscθ=rycosθ=xr secθ=rxtanθ=yx cotθ=xy

Example: The terminal side of an angle, θ, in standard position contains the point (8, -15). Evaluate all six trigonometric functions for this angle.

First, find r. (Note, even though ‘y’ is negative, ‘r’ is always positive.)

r=x2+y2r=64+225r=289r=17

x = 8, y = -15, r = 17

sinθ=−1517 cscθ=−1715cosθ=817 secθ=178tanθ=−158 cotθ=−815 -5-10-1510

- 15r8

Try These: a. The terminal side of an angle θ in standard position contains the point (- 3, 4 ). Evaluate all 6 trigonometric functions for θ.

b. If csc θ = 2 and θ lies in Quadrant 2, then evaluate the five remaining trigonometric functions.

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