Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 67

Capitolo 2, Problema 67

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [sin⁻¹ (− 3/5)]

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Recognize that the expression is \( \tan(\sin^{-1}(-\frac{3}{5})) \). This means we first find an angle \( \theta \) such that \( \sin \theta = -\frac{3}{5} \), and then find \( \tan \theta \).

Draw a right triangle or visualize the angle \( \theta \) in the coordinate plane where \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = -\frac{3}{5} \). Since sine is negative, \( \theta \) is in either the third or fourth quadrant.

Use the Pythagorean theorem to find the adjacent side of the triangle: if opposite side = 3 and hypotenuse = 5, then adjacent side = \( \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \).

Determine the sign of the adjacent side based on the quadrant of \( \theta \). Since sine is negative and cosine (adjacent/hypotenuse) is positive in the fourth quadrant, take adjacent side as positive 4.

Calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}} = \frac{-3}{4} \). This gives the exact value of the expression.

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Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function returns the angle whose sine is a given value. For sin⁻¹(−3/5), it finds an angle θ such that sin(θ) = −3/5. The output angle is typically in the range [−π/2, π/2], which helps determine the correct quadrant for the angle.

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Inverse Sine

Right Triangle Trigonometry

Using a right triangle to represent the angle from the inverse sine helps visualize and calculate other trigonometric ratios. Given sin(θ) = opposite/hypotenuse, you can find the adjacent side using the Pythagorean theorem, enabling the calculation of tan(θ) = opposite/adjacent.

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45-45-90 Triangles

Tangent Function and Its Relationship to Sine and Cosine

Tangent of an angle is the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). After finding sin(θ), you can find cos(θ) using the Pythagorean identity cos²(θ) = 1 − sin²(θ), then compute tan(θ). This relationship is key to finding the exact value of tan[sin⁻¹(−3/5)].

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

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