Section 12-4

Then you applied these to several oblique triangles and developed the law of sines and the law of cosines.

Now we’ll extend the definitions of trigonometric ratios to apply to any size angle.

In this investigation you’ll learn how to calculate the sine, cosine, and tangent of non-acute angles on the coordinate plane.

Procedural Note1. Draw a point on the positive x-axis. Rotate the

point counterclockwise about the origin by the given angle measure and draw the image point. (If the angle measure is negative, rotate clockwise.) Then connect the image point to the origin. The angle between the segment and the positive x-axis, in the direction of rotation, represents the amount of rotation.

2. Use your calculator to find the sine, cosine, and tangent of this angle.

3. Estimate the coordinates of the rotated point.4. Use the distance formula to find the length of

the segment.

2 2

2 1 2 1d x x y y

Follow the Procedure Note for each angle measure given below.

An example is shown for 120°.a. 135°b. 210°c. 270°d. 320°e. 100°

Experiment with the estimated x- and y-coordinates and the segment length to find a way to calculate the sine, cosine, and tangent. (Your values are all estimates, so just try to get close.)

Plot the point (3, 1) and draw a segment from it to the origin. Label as A the angle between the segment and the positive x-axis.

Use your method from Step 2 to find the values of sin A, cos A, and tan A.

What happens when you try using the inverse sin-1 to find the value of A? What happens for cos-1 and tan-1 ?

Now consider the general case of the angle between the positive x-axis and a segment connecting the origin to the point (x, y).

Give definitions for the values of sin θ, cos θ , and tan θ.

Suppose you know that sin B = 0.47. That information isn’t enough to determine

whether B is about 28°, 151°, or even 208°. You need additional information to

determine the measure of angle B.

When you draw the point (3, 1) and make the angle with the origin you notice that

1sin

10A

3cos

10A

1tan

3A

But the calculator gives:

One matches with step 3.

• You can use a graph to find the angle.

• Create a right triangle by drawing a vertical line from the end of the segment to the x-axis. Use right triangle trigonometry to find the measure of the angle with its vertex at the origin.

• The acute angle in this reference triangle, labeled B, is called the reference angle.

• Use the measure of the reference angle to find the angle you are looking for.

• In this case, angle B has measure 18.435°.

• Angle A measures 180° -18.435°, or 161.565°.

• Rotate a point counterclockwise 300° from the positive x-axis. The image point is in Quadrant IV, 60° below the x-axis. The reference angle is 60°.

Find sin 300° without a calculator.

• The sine of a 60° angle is . 32

• The sine of an angle is the y-value divided by the distance from the origin to the point.

• Because the y-value is negative in Quadrant IV,

sin 300° = 32

What measure describes an angle, measured counterclockwise, from the positive x-axis to the ray from the origin through (4, 3)?

The reference angle has measure tan-1 = 3/4 or 36.9°. The graph shows that is more than 180°, so 180° 36.9°, or 216.9°.

A reference triangle constructed with a perpendicular to the x-axis has sides 3-4-5.

The graph shows that is more than 180°, so 180° + 36.9°, or 216.9°.