Question

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \sin 390^{\circ} $$

Short answer

\( \sin 390^{\circ} = \sin 30^{\circ} = \frac{1}{2}\).

Step-by-step solution

- Understand Coterminal Angles

Coterminal angles are angles that differ by a multiple of 360°. They share the same initial and terminal sides. To find a coterminal angle between 0° and 360°, subtract 360° from the given angle if it is greater than 360°.

- Find the Coterminal Angle

Subtract 360° from 390° to find a coterminal angle. a 390° - 360° = 30°.So, 390° is coterminal with 30°.

- Evaluate the Sine Function

The sine of an angle is the same for all of its coterminal angles. Therefore, a\( \sin 390^{\circ} = \sin 30^{\circ}\).From trigonometric functions, we know that a\(\sin 30^{\circ} = \frac{1}{2}\).

Key concepts

sine function

The sine function is a fundamental trigonometric function that describes the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
The sine function repeats every 360°, which means it is periodic.
For any angle \theta, the sine function is written as \(\text{sin}(\theta)\).
Understanding how to compute the sine of both small and large angles is important.
This is because the sine function can have angles beyond \(0°\) and \(360°\) that are equivalent or coterminal angles.
The sine value of an angle and its coterminal angles will always be the same.
For instance, in the solution provided, we found that \(\text{sin}(390°) = \text{sin}(30°)\), and then used the fact that \(\text{sin}(30°) = \frac{1}{2}\).
So, anytime you are working with an angle, check if there is a simpler, coterminal angle to make calculations easier.

angle subtraction

Angle subtraction is a useful technique when working with trigonometric functions.
When an angle exceeds 360°, you can subtract 360° to find a coterminal angle within the standard \range of 0° to 360°.
In the given exercise, we had to find the sine of 390°.
Since 390° is greater than 360°, we subtracted 360° to find a coterminal angle.
\( \text{390° - 360° = 30°} \).
Therefore, we see that the sine of 390° is the same as the sine of 30°.
Identifying these coterminal angles is crucial in simplifying problems and making calculations straightforward.

trigonometric identities

Trigonometric identities are equations that are true for all values of the involved variables.
They help simplify complex trigonometric expressions and solve equations.
The identity \(\text{sin}(A - B) = \text{sin}(A)\text{cos}(B) - \text{cos}(A)\text{sin}(B)\) often helps in breaking down problems.
We also use the basic Pythagorean identities: \(\text{sin}^2(x) + \text{cos}^2(x) = 1\).
Under certain circumstances, we may use angle sum and difference identities to solve problems where straightforward angle subtraction isn't enough.
In our example, we didn't employ angle subtraction identities, but recognizing when and how to use them can be beneficial.
Mastering these identities supports deeper understanding of trigonometry and provides a robust toolbox for even more complex problems.