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Magazine Chapter 1: Problem 11
Decide whether these equations are true or false: (a) \(\frac{\sin \theta}{1-\cos \theta}=\frac{1+\cos \theta}{\sin \theta}\) (b) \(\frac{\sec \theta+\csc \theta}{\tan \theta+\cot \theta}=\sin \theta+\cos \theta\) (c) \(\cos \theta-\sec \theta=\sin \theta \tan \theta\) (d) \(\sin (2 \pi-\theta)=\sin \theta\)
Short Answer
Expert verified
(a) True, (b) True, (c) False, (d) False.
Step by step solution
01
Analyze Equation (a)
Start with the equation \( \frac{\sin \theta}{1-\cos \theta}=\frac{1+\cos \theta}{\sin \theta} \). Cross-multiply to get an equivalent equation: \( \sin^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \). This simplifies to \( \sin^2 \theta = 1 - \cos^2 \theta \). Notice that this is the Pythagorean identity: \( \sin^2 \theta = \sin^2 \theta \). Therefore, equation (a) is true.
02
Analyze Equation (b)
Consider the equation \( \frac{\sec \theta+\csc \theta}{\tan \theta+\cot \theta}=\sin \theta+\cos \theta \). Rewrite the left-hand side in terms of sine and cosine:\( \frac{\frac{1}{\cos \theta}+\frac{1}{\sin \theta}}{\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}} \). This simplifies to \( \frac{\frac{\sin \theta+\cos \theta}{\sin \theta \cos \theta}}{\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}} \).The \( \sin^2 \theta + \cos^2 \theta = 1 \) identity reduces the expression to \( \sin \theta + \cos \theta \), which matches the right-hand side. Thus, equation (b) is true.
03
Analyze Equation (c)
For the equation \( \cos \theta-\sec \theta=\sin \theta \tan \theta \), replace secant and tangent with their definitions:\( \cos \theta - \frac{1}{\cos \theta} = \sin \theta \times \frac{\sin \theta}{\cos \theta} \). This simplifies to \( \cos \theta - \frac{1}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \).This can be rewritten as \( \frac{\cos^2 \theta - 1}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \). Using the identity \( \cos^2 \theta - 1 = -\sin^2 \theta \), this becomes \( \frac{-\sin^2 \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos \theta} \), which is false. Thus, equation (c) is false.
04
Analyze Equation (d)
Examine the equation \( \sin (2 \pi - \theta) = \sin \theta \). Use the trigonometric identity for sine of angle subtraction: \( \sin(2\pi - \theta) = \sin(2\pi)\cos(\theta) - \cos(2\pi)\sin(\theta) \).Since \( \sin(2\pi) = 0 \) and \( \cos(2\pi) = 1 \), the expression simplifies to \( -\sin(\theta) \). Thus, we have \( -\sin \theta = \sin \theta \), which is not true in general for all \( \theta \). Therefore, equation (d) is false.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pythagorean identity
The Pythagorean identity is one of the fundamental trigonometric identities, expressing the relationship between sine and cosine. It states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is derived from the Pythagorean Theorem, as it relates to the unit circle where the radius is 1.
- The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane.
- In the context of the unit circle, \( \sin \theta \) corresponds to the y-coordinate and \( \cos \theta \) corresponds to the x-coordinate.
- This identity shows how the squares of the sine and cosine values of an angle add up to 1, akin to how the squares of the sides of a right triangle equal the square of the hypotenuse.
Trigonometric equations
Trigonometric equations involve trigonometric functions and seek values for which the equation holds true. Solving these equations encompasses utilizing various trigonometric identities and algebraic manipulations to find all valid solutions.
- It often involves simplifying the equation using identities such as the Pythagorean identity, reciprocal identities, and more.
- Cross-multiplying is a technique frequently used when handling equations like fractions to make comparisons between expressions simpler.
- Another step might include factoring expressions or isolating trigonometric functions such as sine or cosine.
Sine and cosine
Sine and cosine are the primary trigonometric functions, representing fundamental relationships in right triangles as well as on the unit circle.
Understanding sine and cosine deeply allows for solving trigonometric equations effectively, as seen in the original step-by-step solutions.
- In terms of right triangles, sine is the ratio of the opposite side to the hypotenuse, and cosine is the ratio of the adjacent side to the hypotenuse.
- On the unit circle, sine and cosine define the y and x coordinates, respectively, of a point corresponding to an angle measured from the positive x-axis.
Understanding sine and cosine deeply allows for solving trigonometric equations effectively, as seen in the original step-by-step solutions.
Angle subtraction identity
The angle subtraction identity for sine is a useful formula in trigonometry, helping to break down the sine of a difference of two angles. It can be expressed as \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \). This identity is essential for expanding and simplifying expressions dealing with angle adjustments.
- The formula is particularly useful in transforming complex angle expressions into more manageable forms.
- It is integral for proving equations or simplifying trigonometric equations, as it handles differences that don't align directly with common angles.
