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O Find the exact six trigonometric functions values of 0. 0 8 7

O Find the exact six trigonometric functions values of 0. 0 8 7

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 1** (20 points) Find the **exact** six trigonometric functions values of \( \theta \). ### Diagram Explanation: The image shows a right triangle with: - An angle labeled \( \theta \) - A hypotenuse measuring 8 units - A base (adjacent to \( \theta \)) measuring 7 units - The other side (opposite to \( \theta \)) can be calculated using the Pythagorean Theorem. ### Solution Steps: 1. **Finding the opposite side:** - Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \) - \( a^2 + 7^2 = 8^2 \) - \( a^2 + 49 = 64 \) - \( a^2 = 15 \) - \( a = \sqrt{15} \) 2. **Trigonometric Functions:** - **Sine (\(\sin\)):** \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{15}}{8}\) - **Cosine (\(\cos\)):** \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{8}\) - **Tangent (\(\tan\)):** \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{15}}{7}\) - **Cosecant (\(\csc\)):** \(\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{8}{\sqrt{15}} = \frac{8\sqrt{15}}{15}\) - **Secant (\(\sec\)):** \(\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{8}{7}\) - **Cotangent (\(\cot\)):** \(\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{7}{\sqrt{15}} = \frac{7\sqrt{15}}{15}\)
Transcribed Image Text:**Problem 1** (20 points) Find the **exact** six trigonometric functions values of \( \theta \). ### Diagram Explanation: The image shows a right triangle with: - An angle labeled \( \theta \) - A hypotenuse measuring 8 units - A base (adjacent to \( \theta \)) measuring 7 units - The other side (opposite to \( \theta \)) can be calculated using the Pythagorean Theorem. ### Solution Steps: 1. **Finding the opposite side:** - Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \) - \( a^2 + 7^2 = 8^2 \) - \( a^2 + 49 = 64 \) - \( a^2 = 15 \) - \( a = \sqrt{15} \) 2. **Trigonometric Functions:** - **Sine (\(\sin\)):** \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{15}}{8}\) - **Cosine (\(\cos\)):** \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{8}\) - **Tangent (\(\tan\)):** \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{15}}{7}\) - **Cosecant (\(\csc\)):** \(\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{8}{\sqrt{15}} = \frac{8\sqrt{15}}{15}\) - **Secant (\(\sec\)):** \(\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{8}{7}\) - **Cotangent (\(\cot\)):** \(\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{7}{\sqrt{15}} = \frac{7\sqrt{15}}{15}\)
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