Certainly! Here are some sample problems and solutions related to trigonometry:
Sample Problem 1: Basic Trigonometric Ratios
Problem: Given a right triangle where one angle \( \theta \) is \( 30^\circ \) and the hypotenuse has a length of 10 units. Find the lengths of the opposite and adjacent sides.
Solution: Using the basic sine and cosine definitions:
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Opposite side: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \implies \sin(30^\circ) = \frac{\text{opposite}}{10} \] Since \( \sin(30^\circ) = \frac{1}{2} \): \[ \frac{1}{2} = \frac{\text{opposite}}{10} \implies \text{opposite} = 5 \text{ units} \]
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Adjacent side: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \implies \cos(30^\circ) = \frac{\text{adjacent}}{10} \] Since \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \): \[ \frac{\sqrt{3}}{2} = \frac{\text{adjacent}}{10} \implies \text{adjacent} = 5\sqrt{3} \text{ units} \approx 8.66 \text{ units} \]
Sample Problem 2: Angle Calculation
Problem: Find the angle \( \theta \) if \( \tan(\theta) = 1 \).
Solution: To find \( \theta \): \[ \tan(\theta) = 1 \implies \theta = \tan^{-1}(1) \] The angles whose tangent is 1 are \( 45^\circ \) and \( 225^\circ \) (in degrees), or \( \frac{\pi}{4} \) and \( \frac{5\pi}{4} \) (in radians).
Sample Problem 3: Law of Sines
Problem: In triangle ABC, \( a = 7 \), \( b = 12 \), and \( A = 30^\circ \). Find angle \( B \).
Solution: Using the Law of Sines: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{7}{\sin(30^\circ)} = \frac{12}{\sin B} \] Since \( \sin(30^\circ) = \frac{1}{2} \): \[ \frac{7}{\frac{1}{2}} = \frac{12}{\sin B} \implies \frac{7 \times 2}{1} = \frac{12}{\sin B} \implies 14 = \frac{12}{\sin B} \] Solving for \( \sin B \): \[ \sin B = \frac{12}{14} = \frac{6}{7} \] Now, find \( B \): \[ B = \sin^{-1}\left( \frac{6}{7} \right) \approx 0.7954 \text{ radians } \approx 45.57^\circ \]
Sample Problem 4: Law of Cosines
Problem: In triangle ABC, if \( a = 10 \), \( b = 7 \), and \( c = 5 \), find angle \( C \).
Solution: Using the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos C \] Substituting the values: \[ 5^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \cos C \] Calculating: \[ 25 = 100 + 49 - 140 \cos C \] \[ 25 = 149 - 140 \cos C \] \[ 140 \cos C = 149 - 25 \implies 140 \cos C = 124 \implies \cos C = \frac{124}{140} = \frac{62}{70} \approx 0.8857 \] Then find \( C \): \[ C = \cos^{-1}(0.8857) \approx 0.474 \text{ radians } \approx 27.17^\circ \]
These are just a few examples of various types of problems within trigonometry. Let me know if you need more examples or specific problems!
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