1. Using Trigonometric Ratios
Trigonometric Identities

2. Reference Angles
When an angle is graphed on the coordinate plane, the positive, acute angle formed by the terminal side and the x-axis is called a reference angle.
They form a bowtie!

3. Graph Angles in the Coordinate Plane
Plot an angle in the coordinate plane
1. Sketch the angle with its initial side on the positive x-axis
2. Construct a right triangle from the terminal side to the x-axis, making the reference angle inside the triangle.

4. Trigonometry Ratios in the Coordinate Plane
When you sketch a right triangle in the coordinate plane, the trig ratios are determined using the sides in relation to the reference angle. The legs are x and y values, and the hypotenuse is r (radius of the circle).

5. Signs of Trigonometry Ratios in the Coordinate Plane
Trigonometric ratios can be written in terms of x, y, and r.
x and y can be positive, negative, or 0.
Trigonometric ratios can be positive, negative, 0, or undefined. Recall the unit circle for the quadrant angles.

6. Find a trig ratio given a trig ratio
1. Make a sketch of a right triangle in the correct quadrant. Be sure that you connect to the x axis and make the reference angle at the origin.
2. Label the sides according to the trig ratio given. Be sure to put any + or - signs on the correct side. [SOHCAHTOA]
3. Find the third side (hypotenuse) using the Pythagorean Theorem. [a² + b² = c²]
4. Now use the trig ratios to find the exact value.
[SOHCAHTOA]

7. EXAMPLE Find cos  and tan  given in quadrant I.
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
Since sin  = opposite/hypotenuse, you label those 2 sides.
4. Calculate the adjacent side using a² + b² = c²

8. EXAMPLE Find sin  and tan  given in quadrant II.
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
Since cos  = adjacent/hypotenuse, you label those 2 sides.
4. Calculate the opposite side using a² + b² = c²

9. EXAMPLE Find sin  and cos  given in quadrant III.
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
Since tan  = opposite/adjacent, you label those 2 sides.
4. Calculate the hypotenuse using a² + b² = c²

10. EXAMPLE Find cos  and tan  given in quadrant 4.
1. Make a sketch of angle
2. Draw a right triangle
3. Label the sides
Since sin  = opposite/hypotenuse, you label those 2 sides.
4. Calculate the adjacent side using a² + b² = c²