Question

Find the exact value of each expression. Do not use a calculator. $$ \sin 30^{\circ} \cdot \tan 60^{\circ} $$

Short answer

\(\frac{\sqrt{3}}{2}\)

Step-by-step solution

Identify Trigonometric Values

Recall the exact trigonometric values of \(30^{\circ}\) and \(60^{\circ}\). \(\sin 30^{\circ} = \frac{1}{2}\) and \(\tan 60^{\circ} = \sqrt{3}\).

Apply the Values

Substitute the exact values into the expression: \(\sin 30^{\circ} \cdot \tan 60^{\circ}\). This becomes \(\frac{1}{2} \cdot \sqrt{3}\).

Simplify the Expression

Simplify the product: \(\frac{1}{2} \cdot \sqrt{3} = \frac{\sqrt{3}}{2}\).

Key concepts

sine function

The sine function, denoted as \(\text{sin}\), is a fundamental trigonometric function. It relates an angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. In more formal terms, for an angle \( \theta \):
\[ \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
For example, \( \text{sin}(30^\text{°}) = \frac{1}{2} \). This means if we have a right triangle with a 30-degree angle, the side opposite this angle is half the length of the hypotenuse.
Understanding the sine function will help solve various trigonometric problems, including the one we are analyzing.

tangent function

The tangent function, denoted as \(\tan\), is also a crucial trigonometric function. It relates an angle of a right triangle to the ratio of the length of the opposite side over the adjacent side. For an angle \( \theta \):
\[ \text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For instance, \( \text{tan}(60^\text{°}) = \text{√3} \). This result comes from looking at an equilateral triangle, where each angle is 60 degrees, and all sides are equal before being divided into two right triangles.
Knowing the tangent function enables you to solve problems involving angles and their side ratios comprehensively.

exact trigonometric values

Exact trigonometric values are specific values for sine, cosine, and tangent functions that are often memorized because they frequently occur in trigonometry problems. For common angles like \(0^\text{°}\), \(30^\text{°}\), \(45^\text{°}\), \(60^\text{°}\), and \(90^\text{°}\), these values are:

• \(\text{sin}(30^\text{°}) = \frac{1}{2} \)
• \(\text{cos}(30^\text{°}) = \frac{\text{√3}}{2} \)
• \(\text{tan}(30^\text{°}) = \frac{\text{√3}}{3} \)
• \(\text{sin}(60^\text{°}) = \frac{\text{√3}}{2} \)
• \(\text{cos}(60^\text{°}) = \frac{1}{2} \)
• \(\text{tan}(60^\text{°}) = \text{√3} \)

By having these values at your fingertips, you can quickly and accurately solve many trigonometric expressions without a calculator.

simplifying expressions

Simplifying trigonometric expressions is the process of making them more manageable or finding their simplest form. This often involves utilizing known trigonometric values and algebraic manipulation.
For example, given the expression \( \text{sin}(30^\text{°}) \times \text{tan}(60^\text{°}) \), you first substitute the exact values:
\[ \text{sin}(30^\text{°}) = \frac{1}{2} \] and \[ \text{tan}(60^\text{°}) = \text{√3} \]
So, the expression becomes:
\[ \frac{1}{2} \times \text{√3} = \frac{\text{√3}}{2} \].
Understanding and practicing how to simplify expressions will greatly enhance your trigonometry skills.