Textbook Question
Solve each equation for x.
arccos x + arctan 1 = 11π/12
593
views
5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 15
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = cos⁻¹ (―1)
Verified step by step guidance1
Recall that the function \( \cos^{-1}(x) \), also known as arccosine, gives the angle \( y \) in the range \( 0 \leq y \leq \pi \) such that \( \cos(y) = x \).
Identify the value inside the arccosine function: here, \( x = -1 \). So we need to find \( y \) such that \( \cos(y) = -1 \).
Think about the unit circle and the cosine values at key angles between \( 0 \) and \( \pi \). Cosine corresponds to the x-coordinate on the unit circle.
Recall that \( \cos(\pi) = -1 \), and since \( \pi \) is within the principal range of arccosine, this is the angle we are looking for.
Therefore, the exact value of \( y = \cos^{-1}(-1) \) is \( y = \pi \).
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, denoted as cos⁻¹ or arccos, returns the angle whose cosine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [0, π] radians (or [0°, 180°]). Understanding this range is crucial for finding exact angle values.
Recommended video:
4:49
Inverse Cosine
Cosine Values of Special Angles
Certain angles have well-known cosine values, such as cos(0) = 1, cos(π/2) = 0, and cos(π) = -1. Recognizing these special angles helps in determining the exact value of inverse trigonometric functions without a calculator.
Recommended video:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Exact Values vs. Approximate Values
In trigonometry, exact values are expressed in terms of well-known constants like π or simple fractions, rather than decimal approximations. For inverse cosine, this means identifying the angle in radians or degrees that precisely corresponds to the given cosine value.
Recommended video:
6:04Example 1
Related Videos
Related Practice