{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Trigonometry Lecture Notes_part2

2 example 60 if 5 sin 13 θ = and θ lies in quadrant

Info iconThis preview shows pages 7–12. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2. Example 60 If 5 sin 13 θ = and θ lies in quadrant II, find the exact value of: a. sin 2 θ b. cos2 θ c. tan 2 θ Example 61 Find the exact value of 2 2tan15 1 tan 15 °- ° Three Forms of the Double-Angle Formula for cos2 θ 2 2 2 2 cos 2 cos sin cos 2 2cos 1 cos 2 1 2sin θ θ θ θ θ θ θ =- =- = - Example 62 Verify the identity: 3 cos3 4cos 3cos θ θ θ =- Power-Reducing Formulas 2 2 2 1 cos2 sin 2 1 cos 2 cos 2 1 cos2 tan 1 cos 2 θ θ θ θ θ θ θ- = + =- = + Example 63 Write an expression for 4 cos θ that does not have powers on the trigonometric functions greater than 1. Example 64 Write an equivalent expression for sin 4 x that does not contain powers of trigonometric functions greater than 1. ( ) 4 2 2 2 1 cos2 1 cos2 sin sin sin 2 2 1 cos2 1 2cos2 1 2cos2 cos 2 2 4 4 3 3cos2 2 4cos2 1 cos2 8 8 x x x x x x x x x x x x-- = = + - + - + = -- + + = = Half-Angle Identities sin x 2 = ± 1 – cos x 2 cos x 2 = ± 1 + cos x 2 tan x 2 = ± 1 – cos x 1 + cos x = sin x 1 + cos x = 1 – cos x sin x where the sign is determined by the quadrant in which x 2 lies. Example 65 Find the exact value of cos112.5 ° Solution Because 112.5° = 225° / 2 , we use the half-angle formula for cos α / 2 with α = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the - sign in the half-angle formula. Example 66 Verify the identity: 1 cos2 tan sin 2 θ θ θ- = Half-Angle Formulas for: 1 cos tan 2 sin sin tan 2 1 cos α α α α α α- = = + Example 67 Verify the identity: tan csc cot 2 α α α =- Example 68 Verify the following identity: 2 (sin cos ) 1 sin 2 θ θ θ- = - Solution: 2 2 2 (sin cos ) sin 2sin cos cos 1 cos2 1 cos2 2sin cos 2 2 2 2sin cos 1 sin 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ- =- +- + = +- =- = - Section 7.5 Product-to-Sum and Sum-to-Product Formulas [ ] [ ] [ ] [ ] 1 sin sin cos( ) cos( ) 2 1 cos cos cos( ) cos( ) 2 1 sin cos sin( ) sin( ) 2 1 cos sin sin( ) sin( ) 2 α β α β α β α β α β α β α β α β α β α β α β α β =-- + =- + + = + +- = +-- Example 69 Express each of the following as a sum or a difference: a. sin8 sin3 x x b. sin 4 cos x x Example 70 Express the following product as a sum or difference: cos3 cos2 x x sin sin 2sin cos 2 2 sin sin 2sin cos...
View Full Document

{[ snackBarMessage ]}

Page7 / 18

2 Example 60 If 5 sin 13 θ = and θ lies in quadrant II...

This preview shows document pages 7 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online