How do you find the exact values of $cos θ$ $costheta$ and $sin θ$ $sintheta$ when $tan θ = 1$ $tantheta=1$ ?

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How do you find the exact values of $cos θ$ $costheta$ and $sin θ$ $sintheta$ when $tan θ = 1$ $tantheta=1$ ?

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Answer 1 · 2016-11-15T21:54:58

$sin (θ) = cos (θ) = \frac{\sqrt{2}}{2}$ $sin(theta) = cos(theta) = sqrt(2)/2$

OR

$sin (θ) = cos (θ) = - \frac{\sqrt{2}}{2}$ $sin(theta) = cos(theta) = -sqrt(2)/2$

Explanation:

Given: $tan (θ) = 1$ $tan(theta) = 1$

Use the identity:

${tan}^{2} (θ) + 1 = {sec}^{2} (θ)$ $tan^2(theta) + 1 = sec^2(theta)$

Substitute $1^{2}$ $1^2$ for ${tan}^{2} (θ)$ $tan^2(theta)$

$1^{2} + 1 = {sec}^{2} (θ)$ $1^2 + 1 = sec^2(theta)$

$2 = {sec}^{2} (θ)$ $2 = sec^2(theta)$

Because $sec (θ) = \frac{1}{cos (θ)}$ $sec(theta) = 1/cos(theta)$ we can change the above equation to:

${cos}^{2} (θ) = \frac{1}{2}$ $cos^2(theta) = 1/2$

$cos (θ) = \pm \frac{1}{\sqrt{2}}$ $cos(theta) = +-1/sqrt(2)$

$cos (θ) = \pm \frac{\sqrt{2}}{2}$ $cos(theta) = +-sqrt(2)/2$

$tan (θ) = \frac{sin (θ)}{cos (θ)} = 1$ $tan(theta) = sin(theta)/cos(theta) = 1$

$sin (θ) = cos (θ)$ $sin(theta) = cos(theta)$

$sin (θ) = \pm \frac{\sqrt{2}}{2}$ $sin(theta) = +-sqrt(2)/2$

Because we are given nothing to determine whether $θ$ $theta$ is in the first of the third quadrant:

$sin (θ) = cos (θ) = \frac{\sqrt{2}}{2}$ $sin(theta) = cos(theta) = sqrt(2)/2$

OR

$sin (θ) = cos (θ) = - \frac{\sqrt{2}}{2}$ $sin(theta) = cos(theta) = -sqrt(2)/2$

Answer 2 · 2016-11-15T22:23:25

$cosθ=sinθ=+-sqrt2/2$

Explanation:

Draw a right isosceles triangle. The angles are 45, 45,90
Tan 45 =1,
Using Pythagoras the sides are in the ratio $1:1:sqrt2$
cos 45=sin45= $1/sqrt2$ = $sqrt2/2$
But then we need to look at the angle in the third quadrant to get the negative result.

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How do you find the exact values of $cos θ$ $costheta$ and $sin θ$ $sintheta$ when $tan θ = 1$ $tantheta=1$ ?

2 Answers

Explanation:

Explanation:

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How do you find the exact values of $cos θ$ $costheta$ and $sin θ$ $sintheta$ when $tan θ = 1$ $tantheta=1$ ?