Question

\(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Find the exact value of each expression if \(\theta=60^{\circ} .\) Do not use a calculator. $$ 2 g(\theta) $$

Short answer

1

Step-by-step solution

- Identify given functions

The given functions are:1. \( f(\theta) = \sin \theta \)2. \( g(\theta) = \cos \theta \)

- Substitute value of \( \theta \)

Substitute \( \theta = 60^{\circ} \) into the function \( g(\theta) \):\( g(60^{\circ}) = \cos 60^{\circ} \)

- Find the value of \( \cos 60^{\circ} \)

By using the unit circle or trigonometric values table:\( \cos 60^{\circ} = \frac{1}{2} \)

- Multiply by 2

Now, multiply the value of \( g(60^{\circ}) \) by 2:\( 2 g(60^{\circ}) = 2 \times \frac{1}{2} \)

- Simplify the result

Simplify the expression:\( 2 \times \frac{1}{2} = 1 \)

Key concepts

Sine Function

The sine function, denoted as \(\f(\theta) = \sin \theta\), is a fundamental trigonometric function. It relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. When working with the unit circle, the sine of an angle \( \theta \) represents the y-coordinate of the point where the terminal side of the angle intersects the circle.
For example, if \( \theta = 60^{\circ} \), \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). Understanding this function helps in solving various problems related to angles and lengths in geometry and trigonometry.

Cosine Function

The cosine function, denoted as \(\f(\theta) = \cos \theta\), complements the sine function and is equally important. It relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse. On the unit circle, the cosine of an angle \( \theta \) represents the x-coordinate of the point where the terminal side of the angle intersects the circle.
For instance, \( \cos 60^{\circ} = \frac{1}{2} \). This value is crucial in many trigonometric calculations, as observed in the given exercise where multiplying \( \cos 60^{\circ} \) by 2 yields 1.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a coordinate system. It is a vital tool in trigonometry for defining the sine, cosine, and tangent functions for all angles.
On the unit circle, the coordinates of any point (x, y) on the circle can be written as (cos \(\theta\), sin \(\theta\)). This representation helps in easily finding the trigonometric values of specific angles like 30°, 45°, 60°, and 90°, which are commonly used in problems. For example, at \( \theta = 60^{\circ} \), the point on the unit circle is \( ( \frac{1}{2}, \frac{\sqrt{3}}{2} ) \).

Trigonometric Values

Trigonometric values are specific values of sine, cosine, and tangent functions at common angles; these include 0°, 30°, 45°, 60°, and 90°. Memorizing these values can greatly simplify calculations in trigonometry.
For instance:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

Using these values, you can quickly solve expressions like in the exercise above where \( 2 \cos 60^{\circ} = 2 \times \frac{1}{2} = 1 \). Practicing these will improve your speed and accuracy in trigonometric problem-solving.