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Magazine Chapter 1: Problem 35
Assuming that \(\sin \alpha=a, \cos \beta=b,\) and \(\tan \gamma=c,\) express the stated quantities in terms of \(a, b,\) and \(c .\) (a) \(\sin (-\alpha)\) (b) \(\cos (-\beta)\) (c) \(\tan (-\gamma)\) (d) \(\sin \left(\frac{\pi}{2}-\alpha\right)\) (e) \(\cos (\pi-\beta)\) (f) \(\sin (\alpha+\pi)\) (g) \(\sin (2 \beta)\) (h) \(\cos (2 \beta)\) (i) \(\sec (\beta+2 \pi)\) (j) \(\csc (\alpha+\pi)\) (k) \(\cot (\gamma+5 \pi)\) (1) \(\sin ^{2}\left(\frac{\beta}{2}\right)\)
Short Answer
Expert verified
(a) \(-a\), (b) \(b\), (c) \(-c\), (d) \(\sqrt{1-a^2}\), (e) \(-b\), (f) \(-a\), (g) \(2b\sqrt{1-b^2}\), (h) \(2b^2-1\), (i) \(1/b\), (j) \(-1/a\), (k) \(1/c\), (l) \((1-b)/2\).
Step by step solution
01
Understanding the Problem
We need to express various trigonometric expressions involving angles \(-\alpha\), \(-\beta\), \(-\gamma\), transformations of these angles, and double angle formulas. We will convert each expression using known trigonometric identities and express everything in terms of given values \(a\), \(b\), and \(c\).
02
Calculate \(\sin (-\alpha)\)
The sine function is odd, meaning \(\sin(-x) = -\sin(x)\). Thus, \(\sin(-\alpha) = -\sin(\alpha) = -a\).
03
Calculate \(\cos(-\beta)\)
The cosine function is even, so \(\cos(-x) = \cos(x)\). Hence, \(\cos(-\beta) = \cos(\beta) = b\).
04
Calculate \(\tan(-\gamma)\)
Tangent is an odd function, thus \(\tan(-x) = -\tan(x)\). Therefore, \(\tan(-\gamma) = -\tan(\gamma) = -c\).
05
Calculate \(\sin(\pi/2 - \alpha)\)
Using the co-function identity, \(\sin(\pi/2 - x) = \cos(x)\). Substituting for \(\alpha\), we have \(\sin(\pi/2 - \alpha) = \cos(\alpha)\). We need to express this in terms of \(a\) using \(\sin^2(\alpha) + \cos^2(\alpha) = 1\), leading to \(\cos(\alpha) = \sqrt{1-a^2}\).
06
Calculate \(\cos(\pi - \beta)\)
Using the identity \(\cos(\pi - x) = -\cos(x)\), it follows that \(\cos(\pi - \beta) = -\cos(\beta) = -b\).
07
Calculate \(\sin(\alpha + \pi)\)
The identity is \(\sin(x + \pi) = -\sin(x)\). Therefore, \(\sin(\alpha + \pi) = -\sin(\alpha) = -a\).
08
Calculate \(\sin(2\beta)\)
The double-angle formula for sine is \(\sin(2x) = 2\sin(x)\cos(x)\). We don't have \(\sin(\beta)\) directly, but we have \(\cos(\beta) = b\). Thus, \(\sin(\beta) = \sqrt{1-b^2}\) using \(\sin^2(\beta) + \cos^2(\beta) = 1\). So, \(\sin(2\beta) = 2\sqrt{1-b^2}b\).
09
Calculate \(\cos(2\beta)\)
The double-angle formula for cosine is \(\cos(2x) = 2\cos^2(x) - 1\). Therefore, \(\cos(2\beta) = 2b^2 - 1\).
10
Calculate \(\sec(\beta + 2\pi)\)
The secant function is periodic with period \(2\pi\). Therefore, \(\sec(x + 2\pi) = \sec(x)\). So, \(\sec(\beta + 2\pi) = \sec(\beta) = \frac{1}{b}\).
11
Calculate \(\csc(\alpha + \pi)\)
The identity is \(\csc(x + \pi) = -\csc(x)\). Since \(\csc(x) = \frac{1}{\sin(x)}\), \(\csc(\alpha + \pi) = -\frac{1}{\sin(\alpha)} = -\frac{1}{a}\).
12
Calculate \(\cot(\gamma + 5\pi)\)
Cotangent is periodic with period \(\pi\), so \(\cot(x + n\pi) = \cot(x)\). Therefore, \(\cot(\gamma + 5\pi) = \cot(\gamma) = \frac{1}{c}\).
13
Calculate \(\sin^2(\frac{\beta}{2})\)
Using the half-angle formula for sine, \(\sin^2(\frac{x}{2}) = \frac{1 - \cos(x)}{2}\). Hence, \(\sin^2(\frac{\beta}{2}) = \frac{1 - \cos(\beta)}{2} = \frac{1 - b}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Functions
Trigonometric functions are fundamental in understanding relationships within a right triangle. There are six primary trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. Each of these functions has a specific role:
- Sine ( \( \sin \)): Measures the ratio of the opposite side to the hypotenuse of a triangle.
- Cosine ( \( \cos \)): Represents the ratio of the adjacent side to the hypotenuse.
- Tangent ( \( \tan \)): Calculated as the ratio of the opposite side to the adjacent side.
- Cotangent ( \( \cot \)): Reciprocal of the tangent function.
- Secant ( \( \sec \)): Reciprocal of the cosine function.
- Cosecant ( \( \csc \)): Reciprocal of the sine function.
Double Angle Formulas
Double angle formulas are extensions of basic trigonometric identities. They allow us to express trigonometric functions of doubled angles using trigonometric functions of a single angle.
- Sine: The double-angle formula for sine is \( \sin(2x) = 2\sin(x)\cos(x) \).
- Cosine: The formula \( \cos(2x) = \cos^2(x) - \sin^2(x) \) simplifies to \( 2\cos^2(x) - 1 \) or \( 1 - 2\sin^2(x) \).
- Tangent: \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \).
Half-Angle Formulas
Half-angle formulas help simplify expressions involving the square root of half angles. These are especially useful when an angle isn't easily derived from known values. Consider the following half-angle formulas:
- Sine: \( \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}} \)
- Cosine: \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}} \)
- Tangent: \( \tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} \) or \( \frac{\sin(x)}{1 + \cos(x)} \)
Periodicity of Trigonometric Functions
Trigonometric functions have unique periodic properties that allow them to repeat values over regular intervals. Understanding these intervals helps in transforming angles to equivalent positions on the unit circle.
- Sine and cosine functions have a period of \( 2\pi \). This means \( \sin(x + 2\pi) = \sin(x) \) and \( \cos(x + 2\pi) = \cos(x) \).
- Tangent and cotangent repeat every \( \pi \), hence \( \tan(x + \pi) = \tan(x) \) and \( \cot(x + \pi) = \cot(x) \).
- Secant and cosecant also repeat every \( 2\pi \), like sine and cosine.
