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Chapter 6: Problem 90

Explain how the double-angle formulas are derived.

Short Answer

Expert verified
The double-angle formulas are derived using the sum of angles formulas for sine and cosine. For sine, \( sin(2a) = 2sin(a)cos(a) \). For cosine, two formulas are derived, \( cos(2a) = cos^2(a) - sin^2(a) \) or \( cos(2a) = 2cos^2(a) - 1 \) or \( cos(2a) = 1 - 2sin^2(a) \). For tangent, \( tan(2a) = 2tan(a)/(1 - tan^2(a)) \).

Step by step solution

01

Derivation of double-angle formula for sine

The double-angle formula for sine can be derived by using the trigonometric identity for sine of sum of two angles which is given by: \( sin(a+b) = sin(a)cos(b) + cos(a)sin(b) \). Setting \( a = b \) in the identity, we get: \( sin(2a) = 2sin(a)cos(a) \), which is the double-angle formula for sine.
02

Derivation of double-angle formula for cosine

Similarly, the double-angle formula for cosine can be derived from the cosine of sum of two angles formula: \( cos(a+b) = cos(a)cos(b) - sin(a)sin(b) \). Setting \( a = b \) in this identity, we get: \( cos(2a) = cos^2(a) - sin^2(a) \). We can also rewrite this formula in terms of sine or cosine using Pythagorean identity \( sin^2(a) + cos^2(a) = 1 \), hence \( cos(2a) = 2cos^2(a) - 1 \) or \( cos(2a) = 1 - 2sin^2(a) \). Both are the double-angle formulas for cosine.
03

Derivation of double-angle formula for tangent

To derive the double-angle formula for tangent, we divide the sine double-angle formula by the cosine double-angle formula, and use the identity \( tan(a) = sin(a)/cos(a) \). So, \( tan(2a) = sin(2a)/cos(2a) = 2tan(a)/(1 - tan^2(a)) \). This is the double angle formula for tangent.

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Most popular questions from this chapter

Write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. $$ \sin \left(\tan ^{-1} x-\sin ^{-1} y\right) $$

The number of hours of daylight in Boston is given by $$ y=3 \sin \left[\frac{2 \pi}{365}(x-79)\right]+12 $$where \(x\) is the number of days after January \(l\). Within a year, when does Boston have 10.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day.

solve each equation on the interval \([0,2 \pi) .\) \(2 \cos ^{3} x+\cos ^{2} x-2 \cos x-1=0\) (Hint: Use factoring by grouping.)

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \sin \left(x+\frac{\pi}{2}\right)=\sin x+\sin \frac{\pi}{2} $$

Use words to describe the formula for each of the following: the tangent of the difference of two angles.
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