Textbook Question
Evaluate each expression without using a calculator.
sin (2 cos⁻¹ (1/5))
809
views
5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 9
Textbook Question
Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)
Verified step by step guidance1
Recognize that the problem asks for the angle \( y \) such that \( \cos(y) = -\frac{\sqrt{2}}{2} \). This means we need to find the angle whose cosine value is \( -\frac{\sqrt{2}}{2} \).
Recall the unit circle values for cosine. The value \( \frac{\sqrt{2}}{2} \) corresponds to angles \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \) in the first and fourth quadrants, respectively. Since the cosine is negative, \( y \) must be in the second or third quadrant where cosine values are negative.
Identify the angles in the second and third quadrants where cosine equals \( -\frac{\sqrt{2}}{2} \). These are \( \frac{3\pi}{4} \) (second quadrant) and \( \frac{5\pi}{4} \) (third quadrant).
Recall that the range of the inverse cosine function \( \cos^{-1}(x) \) is \( [0, \pi] \), which means the output angle \( y \) must lie between 0 and \( \pi \).
Therefore, select the angle within the range \( [0, \pi] \) where \( \cos(y) = -\frac{\sqrt{2}}{2} \), which is \( y = \frac{3\pi}{4} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Cosine Function (cos⁻¹ or arccos)
The inverse cosine function returns the angle whose cosine is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [0, π] radians. Understanding this helps find the angle y such that cos(y) equals the given value.
Recommended video:
4:49
Inverse Cosine
Exact Values of Cosine for Special Angles
Certain angles have well-known cosine values expressed in terms of square roots, such as cos(π/4) = √2/2. Recognizing these exact values allows you to identify the angle corresponding to a given cosine value without a calculator.
Recommended video:
6:04Example 1
Handling Negative Cosine Values
Cosine is negative in the second quadrant (angles between π/2 and π). Since arccos outputs angles in [0, π], a negative cosine value corresponds to an angle in this range, typically π minus the reference angle with positive cosine.
Recommended video:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Related Videos
Related Practice