Using Trigonometric Ratios Trigonometric Identities

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!

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.

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).

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.

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]

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

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

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

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