terminal side The ray where the measurement of an angle stops Initial Side (Ray) positive angles Couterclockwise rotation

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1 You should learn to: LESSON (4.4 & 4.) TRIGONOMETRIC FUNCTIONS OF ANY ANGLE. Use reference angles to evaluate trigonometric functions for angles which form or reference triangles.. Use a calculator to evaluate trigonometric functions for any angle.. Find coterminal angles and their corresponding trig ratios. Terms to know: standard position (for an angle), initial side, terminal side, positive versus negative angles, reference angle, reference triangle, coterminal angles. Section 4. dealt with trigonometric functions for acute angles. This section extends the trigonometric functions to any angle by using reference angles and reference triangles. A discussion of angles (and their measures) in the coordinate plane is an important prerequisite to finding trig ratios for all possible angles. An angle is said to be in standard position in the coordinate plane if it is formed by a rotation from the positive x- axis. initial side The ray where the measurement of an angle starts (For Trig; usually the x-axis) Terminal Side (Ray) Positive Angle (Rotation) terminal side The ray where the measurement of an angle stops Initial Side (Ray) positive angles Couterclockwise rotation negative angles Clockwise rotation Terminal Side (Ray) Negative Angle (Rotation) The reference angle for an angle in standard position is the acute angle formed by its terminal side and the horizontal axis (x-axis Example : Sketch angles in standard position having the following measures, and find their reference angles. a. b. 0 c

2 A reference angle can be used to form a reference triangle whose trig ratios are the same as those for the actual angle. A reference triangle is formed by drawing a vertical segment from the terminal side of the angle to the x-axis, forming a right triangle. For a reference triangle, the hypotenuse is always considered to be positive. A leg is considered to be positive if it is to the right of or above the origin. It is considered to be negative if it is to the left of or below the origin. Example : Build reference triangles, and then find the following trig ratios. Be careful with your signs! a. sin b. sin( 0 ) c. cos 40 cos tan( 0 ) csc 40 tan sec( 0 ) cot 40 Sometimes it is necessary to find trig ratios for angles in standard position which pass through a given point in the coordinate plane. In such cases, you do not need to find the angle measure in order to find the trig ratios for the angle. All you need to do is build a reference triangle and use SOH-CAH-TOA. You may need to use the Pythagorean Theorem to build the reference triangle. Example : Find the three primary trig ratios (sin, cos, and tan) for an angle in standard position, whose terminal side passes through the point (,). a. sin x ( ) x 49 x x b. cos x x c. tan

3 Example 4: Find the values of these trigonometric functions of, if sin = / and cos < 0. csc 4 cos 4 cot 4 If an angle does not have a reference angle of 0, 4, or 60, you can still find its trig ratios (in decimal form) by using a calculator. Example : Use a calculator to find the following: a. sin( 7 ) b. cos6 c. csc d. cot 0 sin( 7 ).887 cos sin.700 tan 0 If angles in standard position share the same terminal side, they are called coterminal angles. The angles in the diagram below have measures of 0, 90, and 0. They are coterminal Coterminal angles have measures which differ by multiples of 60. Thus, you can find angles which are coterminal to a given angle by adding or subtracting 60 as many times as you wish. Example 6: Sketch each angle in standard position, and find one negative and one positive coterminal angle for each. a. 0 b and 480 and there are others and 4 and and there are others

4 Since coterminal angles share the same terminal side, they form the same reference angles (and reference triangles). Thus, they have the same trig ratios. Example 7: Given that sin.0, sin =.777, and sin 7.94 (to 4 decimal place accuracy), find the following without using a calculator. a. sin 4 b. sin( 9 ) c. sin 77 sin 4 sin(60 7 ) sin 7.94 sin( 9) sin( 60 ) sin.0 sin 77 sin(70 ) sin.777 ASSIGNMENT Pages 6-68 (Vocabulary Check -4,, 4, 6, 7, 0,, 0, 04 a-c,, 4, 8) + Pages 94-9 (Vocabulary Check -7, a, 4b, 0, -6, 8, 8, 48,, 4, 6, 66, 7, 77-8, ) + Given cos and cos87.0, find cos( ) and cos807 without using a calculator. Pages (4, 48 a,b,d, 68 (degrees only))

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