Math 150, Fall 008, c Benjamin Aurispa Chapter 7, Continued 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-Angle Formulas Formula for Sine: Formulas for Cosine: Formula for Tangent: sin x = sin x cos x cos x = cos x sin x = 1 sin x = cos x 1 tan x = tan x 1 tan x All of these formulas follow directly from the addition formulas for trig functions from the last section. So, if you remember the addition formulas, you can deduce these if you forget them. Show why the the formulas for cosine are true using the addition formula for cosine and the Pythagorean identities. Example: If sin x = 3 5 and x is in Quadrant III, find sin x, cos x, and tan x. 1
Math 150, Fall 008, c Benjamin Aurispa Formulas for Lowering Powers These definitely will reappear in calculus. sin x = 1 cos x tan x = cos x = 1 cos x 1 + cos x 1 + cos x These formulas follow from the double angle formulas for cosine. Example: Express sin 4 x in terms of the first power of cosine. Half-Angle Formulas sin u = ± 1 cos u tan u = 1 cos u sin u cos u = ± 1 + cos u = sin u 1 + cos u These formulas follow directly from the formulas for lowering powers if you substitute u for x and then take square roots of both sides. (You have to do some simplification to arrive at the half-angle formula for tangent.) The choice of sign depends on which quadrant u lies in. If 0 < u < 90, u is in which Quadrant? If 90 < u < 180, u is in which Quadrant? If 180 < u < 70, u If 70 < u < 360, u is in which Quadrant? is in which Quadrant? Example: Find the exact value of cos 15 using a half-angle formula.
Math 150, Fall 008, c Benjamin Aurispa Example: Given that cot x = 5 and that 70 < x < 360, find sin x, cos x, and tan x. Product-to-Sum Formulas sin u cos v = 1 [sin(u + v) + sin(u v)] cos u sin v = 1 [sin(u + v) sin(u v)] cos u cos v = 1 [cos(u + v) + cos(u v)] sin u sin v = 1 [cos(u v) cos(u + v)] These formulas also follow from the addition formulas. Verify the third Product-to-Sum formula by using addition formulas. Example: Find the value of cos 37.5 sin 7.5. Sum-to-Product Formulas sin x + sin y = sin x + y sin x sin y = cos x + y cos x + cos y = cos x + y cos x cos y = sin x + y cos x y sin x y cos x y sin x y These formulas are found by subsituting u = x+y and v = x y into the Product-to-Sum formulas. 3
Math 150, Fall 008, c Benjamin Aurispa Verify the second Sum-to-Product formula by using a Product-to-Sum formula. Example: Find the value of cos π 1 + cos 5π 1 by using a Sum-to-Product Formula. Verify the identity sin 3x + sin 7x = cot x. cos 3x cos 7x 7.4 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So we cannot find the inverse function of sine, cosine, or tangent unless we restrict the domain to an interval where the function is one-to-one. To find the inverse sine function, we restrict the domain of sine to [ π/, π/] so that we have a one-to-one function. We define the inverse sine function, sin 1 x by sin 1 x = y sin y = x. sin 1 x is the number in the interval [ π/, π/] whose sine is x. sin 1 x has domain and range. The inverse sine function is also called arcsine or arcsin. 4
Math 150, Fall 008, c Benjamin Aurispa Remember that for inverse functions, we have cancellation laws. So, for the sine function sin(sin 1 x) = x for 1 x 1 Examples: sin 1 (sin x) = x for π x π sin 1 3 sin 1 ( 1 ) sin 1 sin 1 (sin π 5 ) sin 1 (sin π 3 ) The inverse cosine function is very similar to the inverse sine function. In order to have an inverse for cosine, we restrict the domain of cosine to the interval [0, π]. The inverse cosine function cos 1 is defined by cos 1 x = y cos y = x. cos 1 x is the number in the interval [0, π] whose cosine is x. cos 1 x has domain and range. The inverse cosine function is also called arccosine or arccos. The same cancellation laws hold for cosine. Examples cos 1 0 cos 1 ( ) cos(cos 1 1 7 ) cos 1 (cos 7π 6 ) 5
Math 150, Fall 008, c Benjamin Aurispa The inverse tangent function is also similar. In order to have an inverse for tangent, we restrict the domain of tangent to the interval ( π/, π/). The inverse tangent function tan 1 is defined by tan 1 x = y tan y = x. tan 1 x is the number in the interval ( π/, π/) whose tangent is x. tan 1 x has domain and range. The inverse tangent function is also called arctangent or arctan. The same cancellation laws hold for tangent. Examples tan 1 ( 3) tan 1 1 tan 1 (tan π 1 ) tan 1 (tan 5π 6 ) In order to compose trig functions with inverse trig functions, the best strategy is to draw a right triangle with the properties you are given. Examples: Evaluate the following expressions. csc(cos 1 7 5 ) sin( tan 1 3 5 ) 6
Math 150, Fall 008, c Benjamin Aurispa cos( cot 1 3 ) Write the following as an algebraic expression in x. sin(tan 1 x) sec(sin 1 x) 7.5 Trigonometric Equations We can solve trigonometric equations just like any other equation. Solve sin x 3 = 0 in the interval [0, π). Since the trig functions are periodic, there are going to be infinitely many solutions, unless we specify a specific interval that we want x to be in. Recall that sine and cosine have period π and tangent has period π. So what do you do? For sine and cosine, solve in [0, π), then add kπ. For tangent, solve in [0, π), then add kπ. Find ALL solutions to the above equation. 7
Math 150, Fall 008, c Benjamin Aurispa Find all solutions to the following equations. sec x = 0 (tan x + 1)( 3 tan x 1) = 0 cos x + sin x = 1 8
Math 150, Fall 008, c Benjamin Aurispa If there is a coefficient in front of x, first solve like above, and then divide by the coefficient LAST. (a) Find all solutions to the equation cos 3x + 1 = 0. (b) Find the solutions just on the interval [0, π). tan 5 x 4 9 tan x 4 = 0 Find all solutions of the equation cos 3x cos x + sin 3x sin x = 3 on the interval [0, π). 9