Evaluating Trigonometric Expression

The problem asks us to find the value of a given trigonometric expression involving standard angles 30° and 60°. The expression is:

\(\frac{2}{3}\)tan² 60° + 3 cos² 30° − 2 sec² 30° − \(\frac{3}{4}\)cot² 60°

To evaluate this expression, we need to know the values of the trigonometric ratios for 30° and 60°.

Key Trigonometric Values for 30° and 60°

Here are the required standard trigonometric values:

Now, let's substitute these values into the given expression.

Step-by-Step Calculation

The expression is \(\frac{2}{3}\)tan² 60° + 3 cos² 30° − 2 sec² 30° − \(\frac{3}{4}\)cot² 60°.

Substitute the values:

• tan 60° = \(\sqrt{3}\), so tan² 60° = \((\sqrt{3})^2 = 3\)
• cos 30° = \(\frac{\sqrt{3}}{2}\), so cos² 30° = \((\frac{\sqrt{3}}{2})^2 = \frac{3}{4}\)
• sec 30° = \(\frac{2}{\sqrt{3}}\), so sec² 30° = \((\frac{2}{\sqrt{3}})^2 = \frac{4}{3}\)
• cot 60° = \(\frac{1}{\sqrt{3}}\), so cot² 60° = \((\frac{1}{\sqrt{3}})^2 = \frac{1}{3}\)

Substitute these squared values back into the expression:

\(\frac{2}{3}(3) + 3(\frac{3}{4}) - 2(\frac{4}{3}) - \frac{3}{4}(\frac{1}{3})\)

Now, simplify each term:

• \(\frac{2}{3}(3) = 2\)
• \(3(\frac{3}{4}) = \frac{9}{4}\)
• \(2(\frac{4}{3}) = \frac{8}{3}\)
• \(\frac{3}{4}(\frac{1}{3}) = \frac{3}{12} = \frac{1}{4}\)

Substitute the simplified terms back into the expression:

\(2 + \frac{9}{4} - \frac{8}{3} - \frac{1}{4}\)

Group the terms with the same denominators:

\(2 + (\frac{9}{4} - \frac{1}{4}) - \frac{8}{3}\)

Calculate the terms in the parenthesis:

\(\frac{9}{4} - \frac{1}{4} = \frac{9-1}{4} = \frac{8}{4} = 2\)

Substitute this back:

\(2 + 2 - \frac{8}{3}\)

\(4 - \frac{8}{3}\)

To subtract the fraction from the whole number, find a common denominator, which is 3:

\(\frac{4 \times 3}{3} - \frac{8}{3}\)

\(\frac{12}{3} - \frac{8}{3}\)

\(\frac{12 - 8}{3}\)

\(\frac{4}{3}\)

Convert the fraction to a decimal:

\(\frac{4}{3} \approx 1.333...\)

Rounding to two decimal places, the value is approximately 1.33.

Revision Table: Standard Angle Trigonometric Values

Additional Information: Trigonometric Identities

This problem uses trigonometric ratios at standard angles. It's also important to remember the reciprocal identities that relate some trigonometric functions:

• sec \(\theta = \frac{1}{\text{cos } \theta}\)
• csc \(\theta = \frac{1}{\text{sin } \theta}\)
• cot \(\theta = \frac{1}{\text{tan } \theta}\) or cot \(\theta = \frac{\text{cos } \theta}{\text{sin } \theta}\)

Also, the square notation like tan² \(\theta\) means \((\text{tan } \theta)^2\), cos² \(\theta\) means \((\text{cos } \theta)^2\), and so on.

Being familiar with these identities and the standard angle values is crucial for solving trigonometric expressions and equations.