Math Problem Statement
Solution
The expression given in the image is:
Let's break this down and solve it step-by-step.
- Step 1: Evaluate
The value of is well-known and equals:
- Step 2: Substitute into the expression
Now, substitute into the given equation:
- Step 3: Simplify the denominator
First, simplify the square of :
So, the denominator becomes:
- Step 4: Simplify the whole expression
Now the expression becomes:
Simplify the multiplication:
- Step 5: Rationalize the denominator
To rationalize the denominator, multiply both the numerator and denominator by :
Thus, the simplified result is:
Would you like further details or have any questions on this? Here are some related questions to explore:
- How do you compute other trigonometric identities like and ?
- How do you rationalize a denominator in general?
- What are other methods for simplifying trigonometric expressions?
- How does the tangent function behave for other common angles like and ?
- Can you solve the same expression for different angles, like or ?
Tip: Remember, it's useful to memorize key trigonometric values for angles like , , and , as they often appear in problems like this one.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Trigonometric Identities
Simplification of Expressions
Formulas
\( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
\( 1 + \tan^2 \theta = \sec^2 \theta \)
Theorems
Tangent Function
Rationalization of Denominators
Suitable Grade Level
Grades 10-12