520 CHAPTER 7 Analytic Trigonometry Equation (10) can be written as θ φ θ φ + = + c a b sin cos cos sin 2 2 or, equivalently, θ φ ( ) + = + c a b sin 2 2 (12) where φ satisfies equation (11). • If > + c a b , 2 2 then θ φ ( ) + > sin 1 or θ φ ( ) + <− sin 1, and equation (12) has no solution. • If ≤ + c a b , 2 2 then the solutions of equation (12) are θ φ θ φ π + = + + = − + − − c a b c a b sin or sin 1 2 2 1 2 2 Once the angle φ is determined by equations (11), the above are the solutions to equation (9). SUMMARY Sum and Difference Formulas • α β α β α β ( ) + = − cos cos cos sin sin • α β α β α β ( ) + = + sin sin cos cos sin • α β α β α β ( ) + = + − tan tan tan 1 tan tan • α β α β α β ( ) − = + cos cos cos sin sin • α β α β α β ( ) − = − sin sin cos cos sin • α β α β α β ( ) − = − + tan tan tan 1 tan tan Now Work PROBLEM 95 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 7.5 Assess Your Understanding 1. The distance d from the point ( ) − 2, 3 to the point ( ) 5, 1 is (pp. 13–16) 2. If θ = sin 4 5 and θ is in quadrant II, then θ = cos . (pp. 419–422) 3. (a) π π ⋅ = sin 4 cos 3 (pp. 40 1–403) (b) π π − = tan 4 sin 6 (pp. 40 1–403) 4. If α π α π = − < < sin 4 5 , 3 2 , then α = cos . (pp. 419–422) 5. Two triangles are if the lengths of two corresponding sides are equal and the angles between the two sides have the same measure. (pp. A16–A19) 6. If = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ P 1 3 , 2 2 3 is a point on the unit circle that corresponds to a real number t , then = t sin , t cos = , and = t tan . (pp. 397–398) Concepts and Vocabulary 7. (a) α β α β ( ) + = cos cos cos α β sin sin (b) α β α β ( ) − = sin sin cos α β cos sin 8. True or False α β α β α β ( ) + = + + sin sin sin 2 sin sin 9. True or False π θ θ ( ) − = cos 2 cos 10. True or False If = = f x x g x x ( ) sin and ( ) cos , then α β α β α β + = − g g g f f ( ) ( ) ( ) ( ) ( ) 11. Multiple Choice Choose the expression that completes the Sum Formula for tangent functions: α β ( ) + = tan . (a) α β α β + − tan tan 1 tan tan (b) α β α β − + tan tan 1 tan tan (c) α β α β + + tan tan 1 tan tan (d) α β α β − − tan tan 1 tan tan 12. Multiple Choice Choose the expression that is equivalent to ° ° + ° ° sin60 cos20 cos60 sin20 (a) cos ° 40 (b) sin ° 40 (c) cos ° 80 (d) sin ° 80
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