Find all values of θbetween 0 and 2πthat satisfy each of the following equations. tanθ=-1

Name: Problem 51, Chapter 13: Calculus with Applications
Uploaded: 2020-02-09T06:39:54-08:00
Duration: 1 min 25 s
Channel: Sherrie Fenner
Description: Find all values of θbetween 0 and 2πthat satisfy each of the following equations. tanθ=-1

step 1 Question 51, where does tangent equal tangent theta equal negative 1? Well I know it's at 45 deg

Find all values of θbetween 0 and 2πthat satisfy each of the...

Key Concepts

Periodic Properties of Tangent

The tangent function is periodic with a period of ?, which means that its values repeat every ? radians. This property allows the general solution of trigonometric equations to be expressed as a principal solution plus integer multiples of ?, simplifying the process of finding all angles that satisfy the equation.

Quadrant Analysis for Trigonometric Functions

Determining the sign of the tangent function in various quadrants is crucial when solving equations. Since tangent is negative where sine and cosine have opposite signs, this typically occurs in the second and fourth quadrants. Understanding quadrant behavior is essential for identifying all correct solutions within a specified interval.

Trigonometric Functions - Tangent

The tangent function relates the sine and cosine of an angle through the ratio tan(?) = sin(?)/cos(?). This function is fundamental in trigonometry for relating the sides of a right triangle, and its properties help in solving equations, such as when the tangent of an angle is set equal to a specific value.

Reference Angles in Trigonometry

A reference angle is the acute angle that an angle makes with the horizontal axis. In solving trigonometric equations, reference angles help in determining the positions of angles that share the same trigonometric values, up to a sign. For instance, when evaluating tan(?) = -1, the reference angle corresponding to tan(?) = 1 is used as a base for finding solutions in different quadrants.

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