How to find all solutions to the following equations in the interval 0 ≤ θ < 2π?

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How to find all solutions to the following equations in the interval 0 ≤ θ < 2π?

Find all solutions to the following equations in the interval 0 ≤ θ < 2π. Give the exact value of the solution(s) where possible. If not possible, round your answer to
4 decimal places.

tan θ − 2 cos θ sin θ = 0

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Answer 1 · 2018-05-16T01:23:22

$0, \frac{π}{4}, \frac{3 π}{4}, π, \frac{5 π}{4}, \frac{7 π}{4}, 2 π$ $0, pi/4, (3pi)/4, pi, (5pi)/4, (7pi)/4, 2pi$

Explanation:

tan t - 2sin t.cos t = 0
$\frac{sin t}{cos t} - 2 sin t . cos t = 0$ $sin t/(cos t) - 2sin t.cos t = 0$
$sin t - 2 sin t . {cos}^{2} t = 0$ $sin t - 2sin t.cos^2 t = 0$
Condition $cos t \neq 0$ $cos t != 0$
$sin t (1 - 2 {cos}^{2} t) = 0$ $sin t( 1 - 2cos^2 t) = 0$
Use trig identity: $1 - 2 {cos}^{2} t = - cos 2 t$ $1 - 2cos^2 t = - cos 2t$
$- sin t . cos 2 t = 0$ $- sin t.cos 2t = 0$
Either factor should be zero.
a. $sin t = 0$ $sin t = 0$ --> $t = k π$ $t = kpi$
$F or (0, 2 π)$ $For (0, 2pi)$ the answers are: $t = 0; t = π$ $t = 0; t = pi$ ; and $t = 2 π$ $t = 2pi$
b. $cos 2 t = 0$ $cos 2t = 0$ --> Unit circle gives 2 solutions for 2t:
$2 t = \frac{π}{2} + 2 k π$ $2t = pi/2 + 2kpi$ , and $2 t = \frac{3 π}{2} + 2 k π$ $2t = (3pi)/2 + 2kpi$
1. $2 t = \frac{π}{2} + 2 k π$ $2t = pi/2 + 2kpi$
$t = \frac{π}{4} + k π$ $t = pi/4 + kpi$
For $(0, 2 π)$ $(0, 2pi)$ , the answers are:
$t = \frac{π}{4}$ $t = pi/4$ , and $t = \frac{π}{4} + π = \frac{5 π}{4}$ $t = pi/4 + pi = (5pi)/4$
2. $2 t = \frac{3 π}{2} + 2 k π$ $2t = (3pi)/2 + 2kpi$
$t = \frac{3 π}{4} + k π$ $t = (3pi)/4 + kpi$
For $(0, 2 π)$ $(0, 2pi)$ , the answers are:
$t = \frac{3 π}{4}$ $t = (3pi)/4$ , and $t = \frac{3 π}{4} + π = \frac{7 π}{4}$ $t = (3pi)/4 + pi = (7pi)/4$

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How to find all solutions to the following equations in the interval 0 ≤ θ < 2π?

Find all solutions to the following equations in the interval 0 ≤ θ < 2π. Give the exact value of the solution(s) where possible. If not possible, round your answer to
4 decimal places.

tan θ − 2 cos θ sin θ = 0

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

How to find all solutions to the following equations in the interval 0 ≤ θ < 2π?

Find all solutions to the following equations in the interval 0 ≤ θ < 2π. Give the exact value of the solution(s) where possible. If not possible, round your answer to 4 decimal places. tan θ − 2 cos θ sin θ = 0

1 Answer

Explanation:

Related questions

Impact of this question

Find all solutions to the following equations in the interval 0 ≤ θ < 2π. Give the exact value of the solution(s) where possible. If not possible, round your answer to
4 decimal places.

tan θ − 2 cos θ sin θ = 0