Objectives Students will learn how to: Describe angles Use radian measure Use degree measure Use angles to model and solve real-life problems

Vocabulary standard position initial side terminal side angle of rotation coterminal angle reference angle

Angle = determined by rotating a ray (half-line) about its endpoint. Initial Side = the starting point of the ray Terminal Side = the position after rotation Vertex = the endpoint of the ray Positive Angles = generated by counterclockwise rotation Negative Angles = generated by clockwise rotation Angle Angle in standard position

Angles are labeled with Greek letters such as (alpha),  (beta), and  (theta). as well as uppercase letters A, B, and C Coterminal(opposite)-have the same initial and terminal sides

One unit of measurement for angles is degrees, which are based on a fraction of a circle. Another unit is called a radian, which is based on the relationship of the radius and arc length of a central angle in a circle. Arc length = radius when  = 1 radian

These are the common angles/Radian. Circumference of Circle:

A 360° rotation is a complete rotation A 360° rotation is a complete rotation. A 180° rotation is one-half of a complete rotation. Remember! For an angle θ in standard position, the reference angle is the positive acute angle formed by the terminal side of θ and the x-axis.

I Do: Drawing Angles in Standard Position Draw an angle with the given measure in standard position. A. 320° B. –110° C. 990° Rotate the terminal side 320° counterclockwise. Rotate the terminal side –110° clockwise. Rotate the terminal side 990° counterclockwise.

Check It Out! You Do!! Draw an angle with the given measure in standard position. A. 210° B. 1020° C. –300° Rotate the terminal side 210° counter-clockwise. Rotate the terminal side 1020° counter-clockwise. Rotate the terminal side 300° clockwise.

I DO: Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle.  = 65° 65° + 360° = 425° Add 360° to find a positive coterminal angle. 65° – 360° = –295° Subtract 360° to find a negative coterminal angle. Angles that measure 425° and –295° are coterminal with a 65° angle.

You DO: Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle.  = 10° 10° + 360° = 370° Subtract 360° to find a positive coterminal angle. 10° – 360° = –350° Subtract a multiple of 360° to find a negative coterminal angle. Angles that measure 370° and –350° are coterminal with a 410° angle.

I Do– Finding Coterminal Angles For the positive angle 13 / 6, subtract 2 to obtain a coterminal angle

You DO – Finding Coterminal Angles cont’d For the negative angle –2 / 3, add 2 to obtain a coterminal angle

I do : Finding Reference Angles Find the measure of the reference angle for each given angle. A.  = 135° B.  = –105° –105° The measure of the reference angle is 45°. The measure of the reference angle is 75°.

You Do: Finding Reference Angles Find the measure of the reference angle for each given angle. C.  = 325° D.  = 105° E.  = –115° 105° –115° The measure of the reference angle is 75° The measure of the reference angle is 35°. The measure of the reference angle is 65°

I DO : Converting Degrees to Radians Convert each measure from degrees to radians. A. 85° 17 85° π radians 180° 36 = 17 π 36 A. 90° 1 90° π radians 180° 2 = π 2

You Do! Example 1 Convert each measure from degrees to radians. A. –36° -1 -36° π radians 180° 5 = - π 5 B. 270° 3 270° π radians 180° 2 = 3π 2

I Do: Converting Radians to Degrees Convert each measure from radians to degrees. 2π 3 A. 2 π 1 3 radians 60 180° Π radians = 120° π 6 B. π 1 6 radians 30 180° Π radians = 30°

You Do! Example 2 Convert each measure from radians to degrees. 5π 6 A. 5 π 1 6 radians 30 180° Π radians = 150° 3π 4 B. - 3 π 1 4 radians 45 180° Π radians = -135° -

Example 4: Finding Values of Trigonometric Functions P (–6, 9) is a point on the terminal side of  in standard position. Find the exact value of the six trigonometric functions for θ. Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

Example 4 Continued Step 2 Find sin θ, cos θ, and tan θ.

Example 4 Continued Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.

Because r is a distance, its value is always positive, regardless of the sign of x and y. Helpful Hint

Check It Out! Example 4 P(–3, 6) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ. y x P(–3, 6)  θ Step 1 Plot point P, and use it to sketch a right triangle and angle θ in standard position. Find r.

Check It Out! Example 4 Continued Step 2 Find sin θ, cos θ, and tan θ.

Check It Out! Example 4 Continued Step 3 Use reciprocals to find csc θ, sec θ, and cot θ.