Domanda del libro di testo
Determine whether each statement is true or false. If false, tell why. csc 22° ≤ csc 68°
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Ch. 2 - Acute Angles and Right Triangles
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 3, Problema 13
Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 6, c = 7
Guida verificata passo dopo passo1
Identify the sides of the right triangle ABC, where the right angle is at C. This means side c is the hypotenuse, and sides a and b are the legs adjacent to angle C. Given a = 6 and c = 7, we need to find side b using the Pythagorean theorem.
Apply the Pythagorean theorem: \(a^2 + b^2 = c^2\). Substitute the known values: \(6^2 + b^2 = 7^2\).
Simplify the equation: \(36 + b^2 = 49\). Then solve for \(b^2\) by subtracting 36 from both sides: \(b^2 = 49 - 36\).
Calculate \(b\) by taking the square root of both sides: \(b = \sqrt{13}\). This gives the exact length of side b.
Next, find the six trigonometric functions for angle B. Recall that angle B is opposite side b, adjacent to side a, and the hypotenuse is c. Use the definitions: \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}\), \(\cos B = \frac{a}{c}\), \(\tan B = \frac{b}{a}\), and their reciprocals \(\csc B = \frac{c}{b}\), \(\sec B = \frac{c}{a}\), \(\cot B = \frac{a}{b}\). Substitute the known values and rationalize denominators where necessary.
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Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (side opposite the right angle) equals the sum of the squares of the other two sides. It is expressed as c² = a² + b². This theorem allows you to find an unknown side length when the other two are known.
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Percorso guidato
5:19
Solving Right Triangles with the Pythagorean Theorem
Trigonometric Functions in Right Triangles
The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—relate the angles of a right triangle to the ratios of its sides. For an angle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. The reciprocal functions are cosecant, secant, and cotangent.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Rationalizing Denominators
Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable radical expression. It is a standard practice to present trigonometric values in simplified, exact form.
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Percorso guidato
2:58Rationalizing Denominators
Pratica correlata
Domanda del libro di testo
Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11
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Domanda del libro di testo
Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. 480° F. 32°
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Domanda del libro di testo
Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°
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Domanda del libro di testo
Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.
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Domanda del libro di testo
Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1.
a = 5, b = 12
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