Unit Circle
Trigonometry - Unit Circle - How it Works - Video
Functions and their Reciprocals
Definitions:
Here we have the definitions of sine, cosine, and tangent and their reciprocal identities. We need to remember SOHCAHTOA to help find the angle on the unit circle.
30 °- 60 °- 90° Triangle and 45° - 45° - 90° Triangle
Here we have the ratios of the special triangles 30°-60°-90° and 45°-45°-90° that we are going to use to find the angle on the unit circle.
Example 1:
Here we have sin(π/3). We write that as sin( 1/3 *π) so we can see the fraction a little easier. We know 0 is on the right side and π on the left. Since 1/3 * π is less than π, we know the angle is going to be on the top half.
So we can divide the top half into three sections. The triangle, we draw will be either in the left section or right section. Since we have 1/3 * π, the hypotenuse will be the first red line after 0.
We have a 30°-60°-90° triangle. Knowing the ratios is important to understand the points. 1/2 is across from 30°; sqrt(3)/2 is across from 60°; 1 is across from 90°. Using SOHCAHTOA, sin60° is equal to opposite over hypotenuse, which is sqrt(3)/2 / 1, and the result of that is sqrt(3)/2.
So sin(π/3) is sqrt(3)/2.
Example 1
Example 1:
Here we have sin(π/3). We write that as sin( 1/3 *π) so we can see the fraction a little easier. We know 0 is on the right side and π on the left. Since 1/3 * π is less than π, we know the angle is going to be on the top half.
So we can divide the top half into three sections. The triangle, we draw will be either in the left section or right section. Since we have 1/3 * π, the hypotenuse will be the first red line after 0.
We have a 30°-60°-90° triangle. Knowing the ratios is important to understand the points. 1/2 is across from 30°; sqrt(3)/2 is across from 60°; 1 is across from 90°. Using SOHCAHTOA, sin60° is equal to opposite over hypotenuse, which is sqrt(3)/2 / 1, and the result of that is sqrt(3)/2.
So sin(π/3) is sqrt(3)/2.
Example 2
Example 2:
Here we have tan(9π/4). We write that as sin( 9/4 *π) so we can see the fraction a little easier. This is time, we have an angle greater than 2π.. Well, let's reduce so we have a more comfortable angle.
2 * π is the same as 8/4 π. So we can subtract 9/4 * π and 8/4 * π, and the result is 1/4 * π. Now that we have the angle, we can cut the circle into sections. We know 0 is on the right side and π on the left. Since 1/4 * π is less than π, we know the angle is going to be on the top half.
So we can divide the top half into four sections. The triangle, we draw will be either in the left section or right section. Since we have 1/4 * π, the hypotenuse will be the first blue line after 0.
We have a 45°-45°-90° triangle. Knowing the ratios is important to understand the points. sqrt(2)/2 is across from 45°; sqrt(2)/2 is across from 45°; 1 is across from 90°. Using SOHCAHTOA, tan45° is equal to opposite over adjacent, which is sqrt(2)/2 / sqrt(2)/2 , and the result of that is 1.
So sin(9π/4) is 1.
Example 3
Example 3:
Here we have cos(7π/6). We write that as sin( 7/6 *π) so we can see the fraction a little easier. We know 0 is on the right side and π on the left. Since 7/6 * π is greater than π, we know the angle is going to be on the bottom half.
So we can divide the top half into six sections. The triangle, we draw will be either in the left section or right section. Since we have 7/6 * π, the hypotenuse will be the first red line after π.
We have a 30°-60°-90° triangle. Knowing the ratios is important to understand the points. 1/2 is across from 30°; sqrt(3)/2 is across from 60°; 1 is across from 90°. Using SOHCAHTOA, cos30° is equal to adjacent over hypotenuse, which is sqrt(3)/2 / 1, and the result of that is sqrt(3)/2.
But we have to be careful here, the numbers that we have used are distances, and distances are always positive. Since we went left on the unit circle, we went left on the coordinate grid as well, which makes x negative, so our answer will be negative.
So cos(7π/6) is -sqrt(3)/2.
Example 4
Example 4:
Here we have sec(2π/3). We write that as sec( 2/3 *π) so we can see the fraction a little easier. We know 0 is on the right side and π on the left. Since 2/3 * π is less than π, we know the angle is going to be on the top half.
So we can divide the top half into three sections. The triangle, we draw will be either in the left section or right section. Since we have 2/3 * π, the hypotenuse will be the second red line after 0.
We have a 30°-60°-90° triangle. Knowing the ratios is important to understand the points. 1/2 is across from 30°; sqrt(3)/2 is across from 60°; 1 is across from 90°. This time we were given the reciprocal of cosine, so we have flip SOHCAHTOA. Since cos30° is equal to adjacent over hypotenuse, sec30° is hypotenuse over adjacent, which is 1 / 1/2, and the result of that is 2.
But we have to be careful here, the numbers that we have used are distances, and distances are always positive. Since we went left on the unit circle, we went left on the coordinate grid as well, which makes x negative, so our answer will be negative.
So sec(2π/3) is -2.