Math Problem Statement

Find tan(-75°) without using a calculator.

Solution

To solve tan⁡(−75∘)\tan(-75^\circ) without a calculator, we use the Sum and Difference Formula for Tangent:

tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}

Step 1: Break down −75∘-75^\circ

We can express −75∘-75^\circ as: −75∘=−45∘−30∘-75^\circ = -45^\circ - 30^\circ

Step 2: Apply the formula

Using the tangent difference formula:

= \frac{\tan(-45^\circ) - \tan(-30^\circ)}{1 + \tan(-45^\circ) \tan(-30^\circ)}$$ ### Step 3: Recall values of tangent - $$\tan(-45^\circ) = -1$$, - $$\tan(-30^\circ) = -\frac{1}{\sqrt{3}}$$. ### Step 4: Substitute the values $$\tan(-75^\circ) = \frac{-1 - \left(-\frac{1}{\sqrt{3}}\right)}{1 + (-1)\left(-\frac{1}{\sqrt{3}}\right)}$$ Simplify the numerator: $$-1 - \left(-\frac{1}{\sqrt{3}}\right) = -1 + \frac{1}{\sqrt{3}}$$ Simplify the denominator: $$1 + (-1)\left(-\frac{1}{\sqrt{3}}\right) = 1 + \frac{1}{\sqrt{3}}$$ $$\tan(-75^\circ) = \frac{-1 + \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}}$$ ### Step 5: Simplify the fraction Multiply numerator and denominator by $$\sqrt{3}$$ to eliminate the square root: $$\tan(-75^\circ) = \frac{-\sqrt{3} + 1}{\sqrt{3} + 1}$$ This is the exact value of $$\tan(-75^\circ)$$. Would you like a detailed step-by-step for further understanding or have any specific questions? Here are related questions for you to explore: --- ### Related Questions: 1. What is the geometric interpretation of negative angles in trigonometry? 2. Can you derive the tangent values for standard angles like $$30^\circ$$ or $$45^\circ$$? 3. How do the sum and difference formulas apply to sine and cosine functions? 4. Why does multiplying by $$\sqrt{3}$$ help in simplifying the fraction? 5. How would you find the exact value of $$\cot(-75^\circ)$$ or $$\sec(-75^\circ)$$? --- ### Tip: Memorize common trigonometric values like $$\tan(30^\circ) = \frac{1}{\sqrt{3}}$$, $$\tan(45^\circ) = 1$$, and their negatives to solve problems quickly without a calculator.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Angle Sum and Difference Identities

Formulas

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Theorems

Sum and Difference Formula for Tangent

Suitable Grade Level

Grades 10-12

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