Angle Exploration ActivityObjectives:

I can locate the terminal side of a given angle in standard position. I can find angles that are coterminal to a given angle. I can identify angles that have the same reference angle. I can find the exact value of trig functions of angles with reference angles 0, 30, 45, 60, 90. I can explain the relationship between trig values of any angle (large or negative), and the trig value of their reference angles. I can solve for the trig function value of an unknown angle given another trig function and the angle’s quadrant.

Key Terms:Angle: The amount of rotation from the initial side to the terminal side of the angle.Standard Position: Initial side of the angle is at 0 (positive x-axis). Counterclockwise movement toward the terminal side gives a positive angle. Clockwise movement toward the terminal side gives a negative angle.Coterminal Angles: Angles that have the same terminal side when placed in standard position.Reference Angle: The positive angle measurement between the x-axis and the terminal side of a standard position angle.

30°

60°90°

150°

180°210°

300°270°

240°

330°

Original by Dr. Lee SingletonNSF: DRL #1623405

360°

120°

0°

1. Find two angles (one positive, one negative) that are coterminal with 310°.

2. What strategy can you use to find an angle between 0 and 360° that is coterminal with an angle you are given?

3. An angle that has a reference angle of 30° has a terminal side located 30° away from the x-axis. Find three angles in each quadrant have a reference angle of 30°. Label them on the figure below.

4. Label the following angles on the figure below, find the quadrant and reference angle for each.a. 245° b. -105° c. 48° d. 310° e. -240° f. 175°g. -75° h. 1035° i. -635° j. 455° k. 730° m. -199°

Trig Definitions for Any AngleYou can use the following definitions to find the trig values of any angle:

Trig values of coterminal angles are always identical. if A and B are coterminal.Trig values of angles with the same reference angle are similar. They only differ by the positive or negative, depending on the quadrants and functions being used. For angle A, and its reference angle Ā, we get: ,

, etc., You can draw a triangle connected to the x-axis to find the trig value of any angle.The reference angle is the angle inside the triangle connected to the origin. Sides can be labeled with appropriate positive and negative values.5. Sketch the reference angle of each angle and use the figure below to find all six trig functions of angles A, B, C, and D.

θ

ry

x

(x,y)

D

C

A

B

(0,-4)(4,-3)

(4,3)(-2,5)

CT

S A

There are a few special angles that allow us to find the exact value of each trig function. These are based on the geometry of an equilateral triangle and a square. Consider the figures below:

6. In each quadrant, draw a triangle with an angle of 30° placed at the origin, and one side connected to the x-axis. Find an angle in each quadrant with a terminal side that ends at the hypotenuse of the triangle you drew in that quadrant. Then label the sides of each triangle and use them to find the exact value of the sin, cos, and tan of these four angles.QI: _________ QII: _________ QIII:_________ QIV: __________QI: QII:

QIII: QIV:

30

12

602

45

452

7. Use the physical triangles given to draw in the terminal sides of the following angles. Draw in the reference angle and find the sin, cos, and tan values of each angle:A) -135° Ref. Angle = _______

B) 420° Ref. Angle = ________

C) 120°

D) 660°

8. If , and in QII, find .Hint:a) Draw a right triangle in QII connected to the x-axis.b) Label the sides based on the given trig function (opp, adj, hyp)Be careful – label negatives appropriately. Hyp is always positive.c) Use the Pythagorean Theorem to find the third side.Again – be careful with labeling using a positive or negative.d) Find the trig function required based on the appropriate ratio of sides.

9. If , and in QIII, find the six trig functions of .

10.If , and , find .

11.If an angle, A, is between 0 and 360°, then what formulas could you use to find Ā, the reference angle of A?QI: Ā = QII: Ā = QIII: Ā = QIV: Ā =