Trigonometric Function Applications

6.6 Trigonometric Function Applications
 
This image is an infographic titled 'Steps to Solve a Trigonometric Equation.' It has a flowchart with four steps, each in a separate box connected by red arrows. 1. The first box says, 'Simplify the equation by using suitable identities if needed.' 2. The second box says, 'Determine the reference angle, using the value of the trigonometric ratio regardless of sign.' 3. The third box says, 'Find the angle in the quadrant that corresponds to the trigonometric ratio sign and range.' 4. The fourth box says, 'Write the solutions obtained.' The background is white, and the text is in dark blue. The logo 'Pandai' is at the bottom right corner.
 
Example
Question

Solve the equation \(\text{sin }\theta = -0.5446\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

Solution

Reference angle:

\(\begin{aligned} \alpha &= \text{sin}^{-1}(0.5446)\\ \alpha &= 33^{\circ}. \end{aligned}\)

\(\text{sin }\theta\) is negative, so \(\theta\) is in the quadrant \(\text{III}\) and \(\text{IV}\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

\(\begin{aligned} \theta&=180^\circ+33^\circ \\ &=213^\circ \end{aligned}\) and \(\begin{aligned} \theta&=360^\circ-33^\circ \\ &=327^\circ. \end{aligned}\)

 

Trigonometric Function Applications

6.6 Trigonometric Function Applications
 
This image is an infographic titled 'Steps to Solve a Trigonometric Equation.' It has a flowchart with four steps, each in a separate box connected by red arrows. 1. The first box says, 'Simplify the equation by using suitable identities if needed.' 2. The second box says, 'Determine the reference angle, using the value of the trigonometric ratio regardless of sign.' 3. The third box says, 'Find the angle in the quadrant that corresponds to the trigonometric ratio sign and range.' 4. The fourth box says, 'Write the solutions obtained.' The background is white, and the text is in dark blue. The logo 'Pandai' is at the bottom right corner.
 
Example
Question

Solve the equation \(\text{sin }\theta = -0.5446\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

Solution

Reference angle:

\(\begin{aligned} \alpha &= \text{sin}^{-1}(0.5446)\\ \alpha &= 33^{\circ}. \end{aligned}\)

\(\text{sin }\theta\) is negative, so \(\theta\) is in the quadrant \(\text{III}\) and \(\text{IV}\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

\(\begin{aligned} \theta&=180^\circ+33^\circ \\ &=213^\circ \end{aligned}\) and \(\begin{aligned} \theta&=360^\circ-33^\circ \\ &=327^\circ. \end{aligned}\)