Textbook Question
Use a half-angle identity to find each exact value.
sin 195°
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.1.34
Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III
Verified step by step guidance1
Recall that the cosecant function is the reciprocal of the sine function, so we can find \(\sin \theta\) by taking the reciprocal of \(\csc \theta\). Write this as \(\sin \theta = \frac{1}{\csc \theta}\).
Calculate \(\sin \theta\) using the given value \(\csc \theta = -\frac{5}{2}\), so \(\sin \theta = -\frac{2}{5}\). Remember the sign is negative because \(\csc \theta\) is negative in quadrant III.
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\). Substitute \(\sin \theta = -\frac{2}{5}\) and solve for \(\cos \theta\).
Determine the sign of \(\cos \theta\) in quadrant III. Since cosine is negative in quadrant III, choose the negative root for \(\cos \theta\).
Find the remaining trigonometric functions using the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Use the values of \(\sin \theta\) and \(\cos \theta\) found in previous steps.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
Each trigonometric function has a reciprocal counterpart: sine and cosecant, cosine and secant, tangent and cotangent. Knowing one function allows you to find its reciprocal by taking the inverse (e.g., csc θ = 1/sin θ). This relationship is essential for determining missing functions when one is given.
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Introduction to Trigonometric Functions
Sign of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant of the angle θ. In quadrant III, both sine and cosine are negative, while tangent is positive. Understanding these sign rules helps correctly assign positive or negative values to the functions when calculating them.
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Pythagorean Identity
The Pythagorean identity, sin²θ + cos²θ = 1, links sine and cosine values. Given one function, you can use this identity to find the other. This is crucial for finding all trigonometric functions when only one is known, especially when combined with quadrant sign information.
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6:25Pythagorean Identities
Related Practice
Textbook Question
Use identities to correctly complete each sentence.
If cos θ = 0.8 and sin θ = 0.6, then tan(-θ) = ________________.
Textbook Question
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1 - 1/sec² x
Textbook Question
Verify that each equation is an identity.
sin² α + tan² α + cos² α = sec² α
1
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Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
sin (3π/4 - x)
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Textbook Question
Find sinθ.
tan θ = -(√7)/2, sec θ > 0