Is arctan 1=220^∘ a true statement? Justify your answer. y=arccos0

Name: Problem 8, Chapter 11: Algebra 2 and Trigonometry
Uploaded: 2019-08-07T18:58:26-08:00
Duration: 1 min 43 s
Channel: Alice .
Description: Is arctan 1=220^∘ a true statement? Justify your answer. y=arccos0

step 1 So in this problem, we're going to be finding the degree measurement of arc cosine of zero. So w

Is arctan 1=220^∘ a true statement? Justify your answer....

Key Concepts

Principal Value Ranges

Each inverse trigonometric function is assigned a specific range known as its principal value range. For instance, the range of the arctan function is typically between -90° and 90° (or -?/2 and ?/2 radians), which means that when evaluating arctan of any number, the result will be confined to this interval. Similarly, the arccos function is defined to yield outputs only within the range of 0° to 180° (or 0 to ? radians).

Periodicity of Trigonometric Functions

Trigonometric functions, like the tangent function, are periodic, meaning that their values repeat after fixed intervals (for tangent, every 180° or ? radians). However, even though multiple angles may yield the same tangent value due to this periodicity, the inverse tangent function returns only the principal value, thereby eliminating the ambiguity of multiple solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine an angle when given a trigonometric ratio. They have well-defined outputs (called principal values) to ensure that each input produces a unique angle. This uniqueness is crucial for solving equations and for practical applications where ambiguity in the solution must be avoided.

The Cosine Function and Its Inverse

The cosine function relates an angle to the ratio of the adjacent side to the hypotenuse in a right triangle or the x-coordinate on the unit circle. Its inverse, the arccos function, is defined to return a unique angle from its principal range (0° to 180° or 0 to ? radians) such that the cosine of that angle equals the given ratio. This allows for the direct determination of an angle from a cosine value while avoiding the many possible solutions that the periodicity of cosine would otherwise produce.

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