Answer:
The measure of the reference angle is 45°.
![\sin(\theta)=\dfrac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Csin%28%5Ctheta%29%3D%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
Step-by-step explanation:
Reference Angle: The <u>smallest possible angle</u> made by the <u>terminal</u> side of the given angle and the <u>x-axis</u>. The reference angle is always between 0° and 90° (0 and π/2 radians).
To convert radians to degrees, multiply by 180/π:
![\implies \sf \theta=\dfrac{3\pi}{4} \cdot \dfrac{180}{\pi}=135^{\circ}](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20%5Ctheta%3D%5Cdfrac%7B3%5Cpi%7D%7B4%7D%20%5Ccdot%20%5Cdfrac%7B180%7D%7B%5Cpi%7D%3D135%5E%7B%5Ccirc%7D)
<u>Positive angles</u> are drawn in a <u>counterclockwise direction</u> from the x-axis.
(See attached for diagram of the angle θ and its reference angle).
<u>Angles on a straight line sum to 180°</u>.
Therefore, to calculate the reference angle, subtract the measure of the angle from 180°:
![\implies \sf Reference\:angle=180^{\circ}-135^{\circ}=45^{\circ}](https://tex.z-dn.net/?f=%5Cimplies%20%5Csf%20Reference%5C%3Aangle%3D180%5E%7B%5Ccirc%7D-135%5E%7B%5Ccirc%7D%3D45%5E%7B%5Ccirc%7D)
The angle 135° lies between 90° and 180° and so is in <u>Quadrant II</u>.
Therefore, the <u>exact trigonometric ratios</u> for 135° are:
![\sin 135^{\circ}=\dfrac{1}{\sqrt{2}} \textsf{ or }\dfrac{\sqrt{2}}{2}\\\\ \textsf{(Since sine function is positive in the Quadrant II)}](https://tex.z-dn.net/?f=%5Csin%20135%5E%7B%5Ccirc%7D%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Ctextsf%7B%20or%20%7D%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5C%20%5Ctextsf%7B%28Since%20sine%20function%20is%20positive%20in%20the%20Quadrant%20II%29%7D)
![\cos 135^{\circ}=-\dfrac{1}{\sqrt{2}} \textsf{ or }-\dfrac{\sqrt{2}}{2}\\\\\textsf{(Since cosine function is negative in the Quadrant II)}](https://tex.z-dn.net/?f=%5Ccos%20135%5E%7B%5Ccirc%7D%3D-%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Ctextsf%7B%20or%20%7D-%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5C%5Ctextsf%7B%28Since%20cosine%20function%20is%20negative%20in%20the%20Quadrant%20II%29%7D)
![\tan135^{\circ}=-1\\\\\textsf{(Since tangent function is negative in the Quadrant II)}](https://tex.z-dn.net/?f=%5Ctan135%5E%7B%5Ccirc%7D%3D-1%5C%5C%5C%5C%5Ctextsf%7B%28Since%20tangent%20function%20is%20negative%20in%20the%20Quadrant%20II%29%7D)
Therefore, the <u>true statements</u> are:
The measure of the reference angle is 45°.
![\sin(\theta)=\dfrac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Csin%28%5Ctheta%29%3D%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)