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In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (3, 7)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 3
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.
Guida verificata passo dopo passo1
Identify the sides of the right triangle relative to angle \( \theta \) at vertex Q. The side opposite \( \theta \) is PR, the side adjacent to \( \theta \) is QR, and the hypotenuse is PQ.
Use the Pythagorean Theorem to find the missing side PR. The theorem states: \( \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \). Substitute the known values: \( 10^2 = PR^2 + 6^2 \).
Rearrange the equation to solve for PR: \( PR^2 = 10^2 - 6^2 \). Then take the square root of both sides to find PR: \( PR = \sqrt{10^2 - 6^2} \).
Once you have the length of PR, calculate the six trigonometric functions of \( \theta \) using the definitions: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \).
Substitute the side lengths into these formulas to express each trigonometric function in terms of the triangle's sides.
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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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6:04Introduction to Trigonometric Functions
Identifying Sides Relative to an Angle
In a right triangle, the side opposite the right angle is the hypotenuse. For a given angle θ, the side directly opposite is the opposite side, and the side next to θ (but not the hypotenuse) is the adjacent side. Correctly identifying these sides is crucial for applying trigonometric functions accurately.
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Percorso guidato
4:18Finding Missing Side Lengths
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