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. $$ g(\theta) $$

Short answer

The value of \(g(60^{\circ})\) is \(\frac{1}{2}\).

Step-by-step solution

Recognize the values of standard trigonometric functions at specific angles

Recall that standard angles, such as \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\) have known sine and cosine values.

Substitute the given angle

Here, \(\theta = 60^{\circ}\). We need to find the cosine of \(60^{\circ}\).

Find the cosine of the given angle

Using the trigonometric table or unit circle, \(\cos 60^{\circ} = \frac{1}{2}\).

State the value of \(g(\theta)\)

Since \(g(\theta) = \cos \theta\), substituting \(\theta = 60^{\circ}\) gives \(g(60^{\circ}) = \frac{1}{2}\).

Key concepts

cosine function

The cosine function is a fundamental trigonometric function, typically denoted as \(\text{cos} \theta\). It computes the horizontal coordinate of a point on the unit circle corresponding to a particular angle \(\theta\). Mathematically, if a right triangle has an angle \(\theta\), the cosine of \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse. Therefore, \(\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\). For example, for \(\theta = 60^{\text{ o}}\), you can use the unit circle or a trigonometric table to find that \(\text{cos} 60^{\text{ o}} = \frac{1}{2}\).Understanding the cosine function is critical in trigonometry because it helps to solve various geometric and real-life problems involving angles and distances.

unit circle

The unit circle is a circle with a radius of 1, centered at the origin \(0,0\) on the coordinate plane. It is instrumental in understanding trigonometric functions as it allows one to easily visualize the sine and cosine values at different angles. Each point \((x, y)\) on the unit circle corresponds to an angle \(\theta\) measured from the positive x-axis, where \(x = \text{cos} \theta\) and \(y = \text{sin} \theta\).For example, when \(\theta = 60^{\text{ o}}\), the coordinates on the unit circle are \(\bigg( \frac{1}{2}, \frac{\text{\textbackslash sqrt}{3}}{2} \bigg)data: \text{\textbackslash frac{ data 1 data 2}\).Thus, \tex{cos} data= data is essential for visualizing how cosine and sine functions behave and finding their values at key angles, like 0°, 30°, 45°, 60°, and 90°.

standard angles

Standard angles are specific angles frequently used in trigonometry, such as 30°, 45°, 60°, and 90°. The trigonometric functions of these angles are well-known and have exact values, making them crucial for solving problems without a calculator. The cosine and sine values for these common angles are as follows:

• 30°: \text{cos} = \frac{\text√3}{2}\(latex\frac{√3}{2}\), \text{sin} = \frac{1}{2}
• 45°: \text{\frac{1}{ \text\sqrt{2}}\textbackslash{frac{value data data}latex\frac{1}{√2}\(, \text{sin} vlaue = \text{ latex \}
• 60°: \text{cos} is ractio latex\frac{1}{2}\), \text{sin}\( latex {latex\}
• 90°: \text{cos} 0\), \text{sin} 1\text<> value \gg
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