Trigonometry Lecture Notes_part2

2 analyze the identity and look for opportunities to

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other side. 2. Analyze the identity and look for opportunities to apply the fundamental identities. Rewriting the more complicated side of the equation in terms of sines and cosines is often helpful. 3. If sums or differences of fractions appear on one side, use the least common denominator and combine the fractions. 4. Don't be afraid to stop and start over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas. Section 7.3 Sum and Difference Formulas
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The Cosine of the Difference of Two Angles cos( ) cos cos sin sin α β α β α β - = + The cosine of the difference of two angles equals the cosine of the first angle times the cosine of the second angle plus the sine of the first angle times the sine of the second angle. Example 49 Use the difference formula for Cosines to find the Exact Value: Find the exact value of cos 15° Solution We know exact values for trigonometric functions of 60° and 45°. Thus, we write 15° as 60° - 45° and use the difference formula for cosines. cos l5° = cos(60° - 45°) = cos 60° cos 45° + sin 60° sin 45° Example 50 Find the exact value of cos 80° cos 20° + sin 80° sin 20°. Example 51 Find the exact value of cos(180º-30º) Example 52 Verify the following identity: cos( ) cot tan sin cos α β α β α β - = + Example 53 Verify the following identity: 5 2 cos (cos sin ) 4 2 x x x π - = - +
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cos( ) cos cos sin sin cos( ) cos cos sin sin sin( ) sin cos cos sin sin( ) sin cos cos sin 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 54 Find the exact value of sin(30º+45º) Example 55 Find the exact value of 7 sin 12 π Example 56 Show that 3 sin cos 2 x x π - =
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Example 57 Find the exact value of tan(105º) Example 58 Verify the identity: tan 1 tan 4 tan 1 x x x π - - = + Example 59 Write the following expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. 7 7 sin cos cos sin 12 12 12 12 π π π π - Section 7.4 Double-Angle and Half-Angle Formulas Double – Angle Formulas tan tan tan( ) 1 tan tan tan tan tan( ) 1 tan tan α β α β α β α β α β α β + + = - - - = +
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2 2 2 sin 2 2sin cos cos 2 cos sin 2tan tan 2 1 tan θ θ θ θ θ θ θ θ θ = = - = - We can derive these by using the sum formulas we learned in section 6.2. 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 cos2 cos sin cos2 2cos 1 cos2 1 2sin θ
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