Domanda del libro di testo
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 1, Problema 1
In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.
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Identify the sides of the right triangle relative to angle \( \theta \) at vertex Q: the side opposite \( \theta \) is \( 3 \), the side adjacent to \( \theta \) is \( 4 \), and the hypotenuse \( PQ \) is unknown.
Use the Pythagorean Theorem to find the hypotenuse \( PQ \): \( PQ^2 = QR^2 + PR^2 \), which translates to \( PQ^2 = 4^2 + 3^2 \).
Calculate \( PQ \) by taking the square root of the sum: \( PQ = \sqrt{4^2 + 3^2} \). This gives the length of the hypotenuse.
Find the six trigonometric functions of \( \theta \) using the side lengths:
- Sine: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{PQ} \)
- Cosine: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{PQ} \)
- Tangent: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} \)
Use the reciprocal identities to find the remaining three functions:
- Cosecant: \( \csc \theta = \frac{1}{\sin \theta} = \frac{PQ}{3} \)
- Secant: \( \sec \theta = \frac{1}{\cos \theta} = \frac{PQ}{4} \)
- Cotangent: \( \cot \theta = \frac{1}{\tan \theta} = \frac{4}{3} \)
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Pythagorean Theorem
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. It is expressed as a² + b² = c². This theorem is essential for finding the missing side length when two sides are known.
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Percorso guidato
5:19
Solving Right Triangles with the Pythagorean Theorem
Right Triangle Trigonometric Functions
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 θ, 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
Identifying Sides Relative to an Angle
In a right triangle, the sides are classified relative to the angle of interest: the opposite side is across from the angle, the adjacent side is next to the angle (but not the hypotenuse), and the hypotenuse is the longest side opposite the right angle. Correctly identifying these sides is crucial for applying trigonometric functions.
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Percorso guidato
4:18Finding Missing Side Lengths
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