Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function.αcos -------2
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
3:41 minutesProblem 58
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:αa. sin ------2𝝅 sec α = ﹣3 , ------ < α < 𝝅2
Verified step by step guidance1
Identify the given information: \( \sec \alpha = -3 \) and \( \frac{\pi}{2} < \alpha < \pi \). This means \( \alpha \) is in the second quadrant.
Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \). Therefore, \( \cos \alpha = -\frac{1}{3} \).
Use the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to find \( \sin \alpha \). Substitute \( \cos \alpha = -\frac{1}{3} \) into the identity.
Solve for \( \sin \alpha \): \( \sin^2 \alpha = 1 - \left(-\frac{1}{3}\right)^2 \). Simplify to find \( \sin \alpha \).
Since \( \alpha \) is in the second quadrant, \( \sin \alpha \) is positive. Use the result from the previous step to find \( \sin \frac{\alpha}{2} \) using the half-angle identity: \( \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and secant, relate the angles of a triangle to the ratios of its sides. Understanding these functions is crucial for solving problems involving angles and their corresponding values. For instance, the sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, while secant is the reciprocal of cosine.
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Quadrants and Angle Ranges
The unit circle is divided into four quadrants, each corresponding to specific ranges of angle values. The given range for α, where π/2 < α < π, indicates that α is in the second quadrant. In this quadrant, sine values are positive, while cosine values are negative, which is essential for determining the correct signs of trigonometric functions when solving for their values.
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Reciprocal Identities
Reciprocal identities in trigonometry express relationships between different trigonometric functions. For example, secant (sec) is the reciprocal of cosine (cos), meaning sec α = 1/cos α. This identity is useful for finding the values of trigonometric functions when given one function, as it allows for the calculation of others based on known values.
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