Question

Find the exact value of each expression. Do not use a calculator. $$ 2 \sin 45^{\circ}+4 \cos 30^{\circ} $$

Short answer

The value is \( \sqrt{2} + 2 \sqrt{3} \)

Step-by-step solution

Evaluate the sine and cosine values

First, find the exact values of \(\sin 45^{\circ}\) and \(\cos 30^{\circ}\). From trigonometric tables or knowledge: \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\).

Substitute the exact values into the expression

Now substitute \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\) into the expression: \[2 \sin 45^{\circ} + 4 \cos 30^{\circ} = 2 \left( \frac{\sqrt{2}}{2} \right) + 4 \left( \frac{\sqrt{3}}{2} \right)\]

Simplify each term

Simplify the terms by performing the multiplication: \[2 \left( \frac{\sqrt{2}}{2} \right) = \sqrt{2}\] \and\ \[4 \left( \frac{\sqrt{3}}{2} \right) = 2\sqrt{3}\]

Combine the simplified terms

Add the simplified terms together: \[ \sqrt{2} + 2\sqrt{3} \]

Key concepts

sine values

Understanding sine values is crucial in trigonometry. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For example, the value of \( \sin 45^{\circ} = \frac{\sqrt{2}}{2}\) because in a 45-degree right triangle, the legs are of equal length, making the hypotenuse \(\sqrt{2}\).
It's important to remember these exact values for key angles such as 30°, 45°, and 60°, as they frequently appear in problems. The more familiar you become with these values, the easier it will be to solve trigonometric problems without a calculator.

cosine values

Cosine values in trigonometry represent the ratio of the adjacent side to the hypotenuse in a right triangle. For instance, \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), as derived from a 30-60-90 triangle where the sides follow the ratio of 1: \(\sqrt{3}: 2\).
You can see these exact trigonometric values used frequently in evaluations and transformations of expressions. It's efficient to memorize these key cosine values, such as \( \cos 0^{\circ} = 1 \), \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), and \( \cos 60^{\circ} = \frac{1}{2} \), to speed up your problem-solving process.

exact trigonometric values

Knowing the exact trigonometric values for specific angles is essential. This knowledge allows you to find the precise value of complex trigonometric expressions without needing a calculator.
For instance, the exact values for sine and cosine of common angles like 0°, 30°, 45°, 60°, and 90° are frequently used. Memorizing the conversion of these angles into their trigonometric values can greatly benefit your mathematical agility. Always ensure to use these exact values to maintain precision in your solutions.

simplifying expressions

Simplifying expressions in trigonometry involves breaking down complex terms into simpler forms.
In the exercise provided, we first found the exact trigonometric values for \(\sin 45^{\circ}\) and \(\cos 30^{\circ}\).
Substituting these known values and then simplifying each term individually allowed us to combine the terms effectively.
This process of substituting and simplifying is fundamental in ensuring that trigonometric expressions are easier to handle and solve.