Problem 43 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \frac{\sin 45^{\circ}}{\cos 45^{\circ}} $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Identify Trigonometric Values

The sine and cosine of 45 degrees are known values. For an angle of 45 degrees, \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 2: Substitute the Trigonometric Values

Substitute the values found in Step 1 into the given expression: \( \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \).

Step 3: Simplify the Fraction

To simplify \( \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \), note that any number divided by itself equals 1. Therefore, the expression simplifies to 1.

Key Concepts

Sine and Cosine Values

In trigonometry, the sine and cosine values for specific angles are constants that you should try to remember. Angles such as 30°, 45°, and 60° often appear in exercises, and their sine and cosine values are widely known. For a 45-degree angle, both

• Sine 45° ( \( \sin 45^{\circ} \) )
• Cosine 45° ( \( \cos 45^{\circ} \) )

share a special property: they are equal to each other. Each of these values can be expressed as \( \frac{\sqrt{2}}{2} \).This means that every time you encounter a 45° angle in trigonometry problems, you can rely on these values. Understanding and memorizing these will make solving similar exercises much easier.Use this trick to make problem-solving more efficient and less memory-intensive. Once you know these key values, half the job is done anytime you work with angles like this.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, especially in trigonometry. To simplify means you make the expression as simple as possible. In the problem provided, you substitute into the fraction the known values of \( \sin 45^{\circ} \) and \( \cos 45^{\circ} \), which are both \( \frac{\sqrt{2}}{2} \).This gives you a fraction in the form:\[\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}.\]A fraction like this seems complicated but remember a key principle: any number divided by itself is 1. So,\[ \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1. \]This is why simplifying fractions often involves looking for components that can cancel each other out. Once identified, the simplification process can become quick and methodical, ultimately leading to the simplified version of the expression.

Trigonometric Ratios

Trigonometric ratios are the foundation of trigonometry. Ratios like sine, cosine, and tangent relate the angles of a triangle to the lengths of its sides. These ratios, though initially abstract, can be visualized through right triangles or the unit circle.In our problem, we used a ratio of sine to cosine, noted as \( \frac{\sin \theta}{\cos \theta} \), which is actually how we define tangent (\( \tan \theta \)). Therefore, if you see \[ \frac{\sin 45^{\circ}}{\cos 45^{\circ}}, \]it's equivalent to \( \tan 45^{\circ} \).This specific ratio simplifies to 1 since the tangent of 45 degrees is 1. This insight helps solve problems quickly—understanding these identities and relationships is crucial. Knowing that you have applied tangent albeit differently, assures you the results are consistent with known trigonometric properties.