Trigonometry / Unit 7: Trigonometric Equations
7.6 Applications and Checking Solutions
Notes
Trig equations model repeating events. Solve for the angle inside, then convert back to time.
- Example: $\displaystyle h(t) = 10 + 8\sin\frac{\pi t}{6}$ gives $h = 14$ when $\displaystyle \sin\frac{\pi t}{6} = \frac{1}{2}$.
- Always keep only solutions in the stated interval (for example $t \ge 0$).
- Check any solution that came from squaring or other steps that can add false answers.
- A sine or cosine can never equal a number beyond $\pm 1$.
Practice
A rider's height is $\displaystyle h(t) = 10 + 8\sin\frac{\pi t}{6}$ feet, with $t$ in seconds. At what time $t \ge 0$ is the height first equal to 14 ft?
$\displaystyle \sin\frac{\pi t}{6} = \frac{1}{2}$, so $\displaystyle \frac{\pi t}{6} = \frac{\pi}{6}$ and $t = 1$.
A rider's height is $\displaystyle h(t) = 10 + 8\sin\frac{\pi t}{6}$ feet. How many times does the height equal 14 ft for $0 \le t < 12$?
$\displaystyle \frac{\pi t}{6}$ is in $[0, 2\pi)$, and $\displaystyle \sin = \frac{1}{2}$ at $\displaystyle \frac{\pi}{6}$ and $\displaystyle \frac{5\pi}{6}$, giving $t = 1$ and $t = 5$.
How many solutions does $\sin x = 1.5$ have?
The sine of any real number is between $-1$ and $1$.
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