Example 1
Completion requirements
| Example 1 |
The point \((-3, 4)\) lies on the terminal arm of an angle in standard position. Determine the primary trigonometric ratios for the angle. Round your answers to the nearest tenth.
Step 1: Sketch the angle and reference triangle.
Step 2: Calculate the length of the line segment between the point and the origin.
\(\begin{align}
r &= \sqrt {\left( {x^2 + y^2 } \right)} \\
r &= \sqrt {\left( {\left( -3 \right)^2 + \left( 4 \right)^2 } \right)} \\
r &= \sqrt {9 + 16} \\
r &= \sqrt {25} \\
r &= 5 \\
\end{align}\)
Step 3: Determine the primary trigonometric ratios for the angle, using the reference triangle.
Step 2: Calculate the length of the line segment between the point and the origin.
\(\begin{align}
r &= \sqrt {\left( {x^2 + y^2 } \right)} \\
r &= \sqrt {\left( {\left( -3 \right)^2 + \left( 4 \right)^2 } \right)} \\
r &= \sqrt {9 + 16} \\
r &= \sqrt {25} \\
r &= 5 \\
\end{align}\)
Step 3: Determine the primary trigonometric ratios for the angle, using the reference triangle.
\(\begin{align}
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{4}{5} \\
\sin \theta &= 0.8 \\
\end{align}\)
\sin \theta &= \frac{y}{r} \\
\sin \theta &= \frac{4}{5} \\
\sin \theta &= 0.8 \\
\end{align}\)
\(\begin{align}
\cos \theta &= \frac{x}{r} \\
\cos \theta &= \frac{{ -3}}{5} \\
\cos \theta &= -0.6 \\
\end{align}\)
\cos \theta &= \frac{x}{r} \\
\cos \theta &= \frac{{ -3}}{5} \\
\cos \theta &= -0.6 \\
\end{align}\)
\(\begin{align}
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{4}{{ - 3}} \\
\tan \theta &\doteq -1.3 \\
\end{align}\)
\tan \theta &= \frac{y}{x} \\
\tan \theta &= \frac{4}{{ - 3}} \\
\tan \theta &\doteq -1.3 \\
\end{align}\)