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Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. See Example 3. cos θ > 0 , sec θ > 0
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 39
Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable.
cos θ = ―5/8 , and θ is in quadrant III
Guida verificata passo dopo passo1
Recall that the six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). We are given cos \( \theta = -\frac{5}{8} \) and that \( \theta \) is in quadrant III.
Since \( \theta \) is in quadrant III, both sine and cosine are negative, and tangent is positive. Use the Pythagorean identity to find sin \( \theta \):
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substitute \( \cos \theta = -\frac{5}{8} \):
\[ \sin^2 \theta + \left(-\frac{5}{8}\right)^2 = 1 \]
Simplify and solve for \( \sin^2 \theta \).
Take the square root of \( \sin^2 \theta \) to find \( \sin \theta \). Since \( \theta \) is in quadrant III, \( \sin \theta \) is negative.
Find \( \tan \theta \) using the definition:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Use the values of \( \sin \theta \) and \( \cos \theta \) found in previous steps.
Calculate the reciprocal functions:
\[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} \]
Remember to rationalize denominators if necessary.
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Trigonometric Functions and Their Relationships
The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are interrelated ratios based on a right triangle or the unit circle. Knowing one function value, such as cosine, allows calculation of others using identities like sin²θ + cos²θ = 1 and definitions like tanθ = sinθ/cosθ.
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Percorso guidato
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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. In quadrant III, both sine and cosine are negative, while tangent is positive. This knowledge helps determine the correct signs of all six functions when given one value and the quadrant.
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6:36Quadratic Formula
Rationalizing Denominators
Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This is often required for final answers in trigonometry to present values in a simplified, standardized form.
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2:58Rationalizing Denominators
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