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Chapter 1 | Functions and Graphs
long does the ramp need to be?
Solution Let x denote the length of the ramp. In the following image, we see that x needs to satisfy the equation sin(10°) = 4/ x . Solving this equation for x , we see that x = 4/sin(10°) ≈ 23.035 ft.
1.19 A house painter wants to lean a 20 -ft ladder against a house. If the angle between the base of the ladder and the ground is to be 60°, how far from the house should she place the base of the ladder?
Trigonometric Identities A trigonometric identity is an equation involving trigonometric functions that is true for all angles θ for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.
Rule: Trigonometric Identities Reciprocal identities
tan θ = sin θ cos θ csc θ = 1 sin θ
θ = cos θ sin θ
cot
sec θ = 1
cos θ
Pythagorean identities
sin 2 θ +cos 2 θ =1 1+tan 2 θ = sec 2 θ
1+cot 2 θ =csc 2 θ
Addition and subtraction formulas
sin ⎛ ⎠ = sin α cos β ±cos α sin β cos( α ± β ) =cos α cos β ∓sin α sin β ⎝ α ± β ⎞
Double-angle formulas
sin(2 θ ) =2sin θ cos θ cos(2 θ ) =2cos 2 θ −1=1−2sin 2 θ =cos 2 θ −sin 2 θ
Example 1.25 Solving Trigonometric Equations
For each of the following equations, use a trigonometric identity to find all solutions. a. 1+cos(2 θ ) =cos θ b. sin(2 θ ) = tan θ
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