Textbook Question
Find the exact value of the variables in each figure.
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
2:23 minutesProblem 2.R.36
Textbook Question
Find a value of θ, in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. csc θ = 9.5670466
Verified step by step guidance1
Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\). This means that \(\sin \theta = \frac{1}{\csc \theta}\).
Substitute the given value of \(\csc \theta\) into the equation: \(\sin \theta = \frac{1}{9.5670466}\).
Calculate the value of \(\sin \theta\) using the reciprocal of \$9.5670466$ (do not compute the final decimal here, just set up the expression).
Use the inverse sine function to find \(\theta\): \(\theta = \sin^{-1} \left( \frac{1}{9.5670466} \right)\).
Since \(\theta\) must be in the interval \([0^\circ, 90^\circ)\), the principal value from the inverse sine function will be the solution. Express \(\theta\) in decimal degrees to six decimal places.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
The cosecant function (csc θ) is the reciprocal of the sine function, defined as csc θ = 1/sin θ. Understanding this relationship allows you to convert between csc θ and sin θ, which is essential for solving equations involving csc θ.
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Inverse Trigonometric Functions
Inverse trigonometric functions, such as arcsin, are used to find the angle θ when the value of a trigonometric function is known. After finding sin θ from csc θ, you use arcsin to determine θ within the specified interval.
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Domain and Range Restrictions
When solving trigonometric equations, it is important to consider the domain and range of the angle θ. Here, θ is restricted to [0°, 90°), meaning the solution must be an acute angle, which affects the choice of the correct inverse function value.
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