Textbook Question
Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.RE.16
Textbook Question
Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using
a. difference identities
b. half-angle identities.
Verified step by step guidance1
Identify the angle \( x = \frac{\pi}{12} \) radians, which is equivalent to 15 degrees. Recognize that \( \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \), so you can use difference identities with angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \).
For part (a), apply the difference identities for sine, cosine, and tangent:
- \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)
- \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)
- \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \)
Use \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \) and substitute the exact values of sine, cosine, and tangent for these standard angles.
For part (b), use the half-angle identities to find \( \sin \frac{\theta}{2} \), \( \cos \frac{\theta}{2} \), and \( \tan \frac{\theta}{2} \) where \( \theta = \frac{\pi}{6} \) because \( \frac{\pi}{12} = \frac{1}{2} \times \frac{\pi}{6} \). The half-angle formulas are:
- \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)
- \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
- \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \) or \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \)
Determine the correct signs for the half-angle values based on the quadrant where \( x = \frac{\pi}{12} \) lies (first quadrant, so sine, cosine, and tangent are positive). Substitute the exact value of \( \cos \frac{\pi}{6} \) into the half-angle formulas.
Simplify the expressions under the square roots and the fractions to get the exact values of \( \sin \frac{\pi}{12} \), \( \cos \frac{\pi}{12} \), and \( \tan \frac{\pi}{12} \) using both methods, without calculating decimal approximations.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Difference Identities
Difference identities express the sine, cosine, or tangent of a difference of two angles in terms of the sines and cosines of the individual angles. For example, sin(a - b) = sin a cos b - cos a sin b. These identities allow exact evaluation of trigonometric functions for angles like π/12 by rewriting them as differences of known angles.
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Half-Angle Identities
Half-angle identities provide formulas to find the sine, cosine, or tangent of half an angle using the cosine or sine of the original angle. For instance, sin(θ/2) = ±√((1 - cos θ)/2). These identities help compute exact trigonometric values for angles like π/12 by relating them to angles like π/6.
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Exact Values of Common Angles
Knowing the exact sine, cosine, and tangent values of standard angles such as π/6, π/4, and π/3 is essential. These values serve as building blocks when applying difference or half-angle identities to find exact values for non-standard angles like π/12.
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