Question

Find the exact value of each expression. Do not use a calculator. $$ 6 \tan 45^{\circ}-8 \cos 60^{\circ} $$

Short answer

2

Step-by-step solution

Evaluate \( \tan 45^{\circ} \)

The \( \tan 45^{\circ} \) is equal to 1, based on the trigonometric values for common angles.Thus, \( \tan 45^{\circ} = 1. \)

Multiply by 6

We need to multiply \( \tan 45^{\circ} \) by 6.\[ 6 \times \tan 45^{\circ} = 6 \times 1 = 6 \]

Evaluate \( \cos 60^{\circ} \)

The \( \cos 60^{\circ} \) is equal to \( \frac{1}{2} \), based on trigonometric values for common angles.Thus, \( \cos 60^{\circ} = \frac{1}{2} \)

Multiply by 8

We need to multiply \( \cos 60^{\circ} \) by 8.\[ 8 \times \cos 60^{\circ} = 8 \times \frac{1}{2} = 4 \]

Subtract the Results

Now, subtract the result of \( 8 \cos 60^{\circ} \) from the result of \( 6 \tan 45^{\circ} \).\[ 6 - 4 = 2 \]

Key concepts

tangent function

The tangent function, denoted as \( \tan \), is one of the primary trigonometric functions, alongside sine and cosine. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is written as:
\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
For the angle \( 45^{\text{°}} \), the tangent function has a special property where:
\( \tan(45^{\text{°}}) = 1 \).
Knowing the values of tangent for common angles like 0°, 30°, 45°, 60°, and 90° helps in quickly solving many trigonometric problems. Here, in our problem, we multiply the tangent of 45 degrees by 6, which means:
\( 6 \times \tan(45^{\text{°}}) = 6 \times 1 = 6 \).

cosine function

The cosine function, denoted as \( \text{cos} \), is another primary trigonometric function. For an angle in a right-angled triangle, the cosine of the angle is the ratio of the length of the adjacent side to the length of the hypotenuse. This can be expressed as:
\( \text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
For the angle \( 60^{\text{°}} \), the cosine value is:
\( \text{cos}(60^{\text{°}}) = \frac{1}{2} \).
In the given problem, we need to multiply the cosine of 60 degrees by 8, which results in:
\( 8 \times \text{cos}(60^{\text{°}}) = 8 \times \frac{1}{2} = 4 \).
Understanding these values and how to manipulate them helps break down complex trigonometric expressions.

common angles

Common angles in trigonometry include 0°, 30°, 45°, 60°, and 90°. These angles often have well-known sine, cosine, and tangent values that are frequently used in various problems. Some important values to remember include:

• \( \tan(45^{\text{°}}) = 1 \)
• \( \text{cos}(60^{\text{°}}) = \frac{1}{2} \)
• \( \text{sin}(30^{\text{°}}) = \frac{1}{2} \)
• \( \text{cos}(30^{\text{°}}) = \frac{\text{√3}}{2} \)
• \( \text{sin}(90^{\text{°}}) = 1 \)

These values are the building blocks for solving many trigonometric problems without needing a calculator.
In our exercise, having known values for \( \tan(45^{\text{°}}) = 1 \) and \( \text{cos}(60^{\text{°}}) = \frac{1}{2} \), allowed us to find the exact value of the expression 6 \( \tan 45^{\text{°}}) - 8 \text{cos} 60^{\text{°}} \) without needing any complex calculations.
First, we computed \( 6 \times 1 = 6 \). Then, we computed \( 8 \times \frac{1}{2} = 4 \). Finally, we subtracted 4 from 6, resulting in 2.