Question

In Problems 19-30, find the exact value of each expression. Do not use a calculator. $$ 4 \cos 45^{\circ}-2 \sin 45^{\circ} $$

Short answer

\backslash sqrt{2}

Step-by-step solution

Identify the Trigonometric Values

Find the exact values of \(\text{cos} 45^{\circ}\) and \( \text{sin} 45^{\circ} \). Recall that \( \text{cos} 45^{\circ} = \frac{\backslash sqrt{2}}{2} \) and \( \text{sin} 45^{\circ} = \frac{\backslash sqrt{2}}{2} \).

Substitute the Values

Plug in the values of \( \text{cos} 45^{\circ} \) and \( \text{sin} 45^{\circ} \) into the expression. This gives us: \( 4 \text{cos} 45^{\circ} - 2 \text{sin} 45^{\circ} = 4 \frac{\backslash sqrt{2}}{2} - 2 \frac{\backslash sqrt{2}}{2} \).

Simplify the Expression

Simplify each term: \( 4 \frac{\backslash sqrt{2}}{2} = 2\backslash sqrt{2} \) and \( -2 \frac{\backslash sqrt{2}}{2} = -\backslash sqrt{2} \). So the expression becomes \( 2\backslash sqrt{2} - \backslash sqrt{2} \).

Combine Like Terms

Combine \( 2\backslash sqrt{2} - \backslash sqrt{2} \) to get \( \backslash sqrt{2} \).

Key concepts

Exact Values of Trigonometric Functions

When working with trigonometric functions, it is essential to know the exact values of certain key angles, such as 0°, 30°, 45°, 60°, and 90°. These angles frequently appear in trigonometric problems, and knowing their exact values can make solving these problems much easier.
For example, the exact values for the cosine and sine of 45° are both \(\frac{\backslash sqrt{2}}{2}\). These values have been derived from the unit circle, a valuable tool in trigonometry.
Memorizing these values can help you find solutions quicker without needing a calculator.
Here are some key trigonometric values to remember:

• \(\text{sin} 0^{\backslash circ} = 0\)
• \(\text{cos} 0^{\backslash circ} = 1\)
• \(\text{sin} 30^{\backslash circ} = \frac{1}{2}\)
• \(\text{cos} 30^{\backslash circ} = \frac{\backslash sqrt{3}}{2}\)
• \(\text{sin} 45^{\backslash circ} = \frac{\backslash sqrt{2}}{2}\)
• \(\text{cos} 45^{\backslash circ} = \frac{\backslash sqrt{2}}{2}\)
• \(\text{sin} 60^{\backslash circ} = \frac{\backslash sqrt{3}}{2}\)
• \(\text{cos} 60^{\backslash circ} = \frac{1}{2}\)
• \(\text{sin} 90^{\backslash circ} = 1\)
• \(\text{cos} 90^{\backslash circ} = 0\)

So whenever you encounter problems that involve these specific angles, quickly use these exact values.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves substituting known values and then performing basic arithmetic operations. For example, in the original exercise, we needed to simplify the expression \(4 \text{cos} 45^{\backslash circ} - 2 \text{sin} 45^{\backslash circ}\).
The process becomes straightforward once you substitute the known values. Since \( \text{cos} 45^{\backslash circ} = \frac{\backslash sqrt{2}}{2}\) and \( \text{sin} 45^{\backslash circ} = \frac{\backslash sqrt{2}}{2}\), we plug these in to get:
\(4 \frac{\backslash sqrt{2}}{2} - 2 \frac{\backslash sqrt{2}}{2}\).
After substitution, the next step is to perform the multiplication for each term:
- \(4 \frac{\backslash sqrt{2}}{2} = 2 \backslash sqrt{2}\)
- \(-2 \frac{\backslash sqrt{2}}{2} = - \backslash sqrt{2}\)
This leads us to a simplified expression of \(2 \backslash sqrt{2} - \backslash sqrt{2}\).
Remember, by knowing the exact values and following the substitution, you can simplify even complex-looking trigonometric expressions step by step without struggle.

Combining Like Terms

Combining like terms is an essential skill you need to simplify algebraic expressions. This principle also applies to trigonometric expressions. Like terms are terms that have the same variable parts (including the trigonometric functions) and can thus be combined by adding or subtracting their coefficients.
In our example, once we simplified the expression to \(2 \backslash sqrt{2} - \backslash sqrt{2}\), we notice both terms have the same radical part \( \backslash sqrt{2}\). This makes them like terms.
To combine them, you simply perform the arithmetic operation on the coefficients. Here it is: \(2 \backslash sqrt{2} - \backslash sqrt{2} = (2-1)\backslash sqrt{2} = \backslash sqrt{2}\).
By following this step, you have combined like terms to further simplify your expression. Always look for such opportunities to make your math problems easier to solve.