Question

How to find exact value of trig functions

Answer 1

There are many different ways to find exact values of trigonometric functions. Some of those include the following:

Using trigonometric identities: A trigonometric identity is a relationship between trigonometric functions that we can use to find an exact value of a trigonmetric function.Using the unit circle: The unit circle is a circle with its center at the origin and a radius of 1 unit. If we place an angle, θ, with its initial side along the positive x-axis, then the point where the angle's terminal side intersects the unit circle is (cos(θ), sin(θ)).Using reference angles: Reference angles are the smallest angle that the terminal side of an angle makes with the x-axis. As it turns out, trigonometric functions of an angle and its reference angle have the same value, except that they may have different signs.

For example, suppose we want to find the exact value of tan(405°). First, we take a look at the angle 405° on a graph with the unit circle:
in this that the reference angle of 405° is 45°. We can use the fact that tangent is positive in the first quadrant and the rule for reference angles to deduce that tan(405°) = tan(45°). Then we can use the well-known fact that tan(45°) = 1 to get that tan(405°) = 1.

We could also use the unit circle along with a trigonometric identity to find the exact value of tan(405°). In trigonometry, a well-known trigonometric identity that relates tangent, sine, and cosine is as follows:

tan(θ)=sin(θ)cos(θ)tan(θ)=sin(θ)cos(θ)

Notice that the terminal side of our 405° angle intersects the unit circle at the point (√22,√22)(22,22). By our unit circle rule, this gives the following:

sin(405o)=√22cos(405o)=√22sin(405o)=22cos(405o)=22

Plugging these values into our trigonometric identity gives the following:

tan(405o)=sin(405o)cos(405o)=√22√22=1tan(405o)=sin(405o)cos(405o)=2222=1

Once again, we get that the exact value of tan(405°) is 1, and we see some of the different ways that we can find exact values of trigonometric functions.