For Most Angles, the Values of the Sine, Cosine, and Tangent Functions Are Non-Repeating

Honors Geometry

Advanced Trigonometry

4. Trigonometric Functions with Exact Values

For most angles, the values of the sine, cosine, and tangent functions are non-repeating decimals which must be approximated by rounding.

For example, sin 40 = 0.642787609…

This is usually rounded to 0.6428 in places such as the trig tables in the back of your Geometry book.

However, some angles have exact values. All the quadrant angles (0, 90, 180, 270) and the special angles (30, 45, 60) have exact values for their sines, cosines, and tangents.

Naturally, any angle which is co-terminal with the angles mentioned above also has exact values for trig functions. For example, 450 and -180 have trig functions with exact values because they are co-terminal with 90 and 180.

In addition, any angle for which a 30, 45, 60 angle is the reference angle has a sine, cosine, and tangent with exact values. For example, 135 has exact trig values because its reference angle is the special angle 45.

We can use any size circle we wish in trigonometry, but the most convenient circle is the unit circle, which has a radius of one unit and a center at the origin.

Sin θ =

Cos θ =

So, for any point on the unit circle, the x co-ordinate represents the cosine of the angle terminating there, and the y-co-ordinate represents the sine. For example, if an angle on the unit circle terminates at (-0.6, 0.8), its cosine is

-0.6, and its sin is 0.8. The tangent is still y/x, so it would be 0.8/-0.6, or -4/3.

For example, a 180 angle terminates at (-1,0). So its cosine is -1, its sine is 0, and its tangent (0/-1) is 0.

Here you can see that the sine of 30 is , the cosine is , and the tangent is. (  = = )

The circle above shows that the sine and cosine of 45 each equal . The tangent of 45 equals 1 (a very useful fact down the road).

You could continue constructing triangles to show that sin 60 = ,

cos 60 = ½ , and tan 60 = .

Sine, cosine, and tangent of other angles

Sin 630

By co-terminal angles,

sin 630 = sin 270 = -1

Cos 240

The reference angle for 240 is 60.

Cosine is negative in Quad III, so

cos 240 = -cos 60 = - ½

Tan -225

By co-terminal angles,

tan(-225) = tan 135

The reference angle for 135 is 45.

Tangent is negative in Quad II, so

tan(-225) = tan 135 = -tan 45 = -1

Any angle whose measure is evenly divisible by 15 has exact values for sine, cosine, and tangent.

Exercises

Fill in – then memorize – the table.

Give exact values for the quantity.

1. sin(-90)12. tan 150

2. cos 90013. sin 150

3. tan(-720)14. cos 300

4. sin 81015. tan 210

5. cos(-270)16. sin 120

6. tan 45017. cos 135

7. sin 13518. tan 225

8. cos 12019. sin 660

9. tan 33020. cos(-45)

10. sin 21021. tan -315

11. cos 315