Example 3
Completion requirements
| Example 3 |
Determine the exact values of the trigonometric ratios for the angle \(210^\circ\).
Step 1: Determine the reference angle, \(\theta_R\).
The angle \(210^\circ\) terminates in Quadrant III because it is greater than \(180^\circ\) and less than \(270^\circ\). To find the reference angle, subtract \(180^\circ\) from \(210^\circ\).
\(\begin{align}
\theta - 180^\circ &= \theta _R \\
210^\circ - 180^\circ &= \theta _R \\
30^\circ &= \theta _R \\
\end{align}\)
Step 2: Determine which ratios are positive and which are negative in Quadrant III.
Using CAST, only tangent is positive in Quadrant III.
Step 3: Using the reference angle, determine the exact values for the angle.
Referring to the special right triangle with an angle of \(30^\circ\), the primary trigonometric ratios for \(30^\circ\) are:
\(\begin{align}
\sin 30^\circ &= \frac{1}{2} \\
\cos 30^\circ &= \frac{{\sqrt 3 }}{2} \\
\tan 30^\circ &= \frac{{\sqrt 3 }}{3} \\
\end{align}\)
To determine the exact values for the angle \(210^\circ\), apply the CAST rule.
\(\begin{align}
\sin 30^\circ &= -\frac{1}{2} \\
\cos 30^\circ &= -\frac{{\sqrt 3 }}{2} \\
\tan 30^\circ &= \frac{{\sqrt 3 }}{3} \\
\end{align}\)
Step 1: Determine the reference angle, \(\theta_R\).
The angle \(210^\circ\) terminates in Quadrant III because it is greater than \(180^\circ\) and less than \(270^\circ\). To find the reference angle, subtract \(180^\circ\) from \(210^\circ\).
\(\begin{align}
\theta - 180^\circ &= \theta _R \\
210^\circ - 180^\circ &= \theta _R \\
30^\circ &= \theta _R \\
\end{align}\)
Step 2: Determine which ratios are positive and which are negative in Quadrant III.
Using CAST, only tangent is positive in Quadrant III.
Step 3: Using the reference angle, determine the exact values for the angle.
Referring to the special right triangle with an angle of \(30^\circ\), the primary trigonometric ratios for \(30^\circ\) are:
\(\begin{align}
\sin 30^\circ &= \frac{1}{2} \\
\cos 30^\circ &= \frac{{\sqrt 3 }}{2} \\
\tan 30^\circ &= \frac{{\sqrt 3 }}{3} \\
\end{align}\)
To determine the exact values for the angle \(210^\circ\), apply the CAST rule.
\(\begin{align}
\sin 30^\circ &= -\frac{1}{2} \\
\cos 30^\circ &= -\frac{{\sqrt 3 }}{2} \\
\tan 30^\circ &= \frac{{\sqrt 3 }}{3} \\
\end{align}\)