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![Find
tan((19 pi)/(12)) exactly using an angle addition or subtraction formula.
[Hint: This diagram of special trigonometry. values may help.].
Choose 1 answer:
(A)
(3-sqrt3)/(3+sqrt3)
(B)
(3-sqrt3)/(3-sqrt3)
(c)
(-3+sqrt3)/(3+sqrt3)
(D)
(3+sqrt3)/(-3+sqrt3)](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/PreCalculus/Trigonometry/Using%20trigonometric%20identities/Q-12.png)
Find exactly using an angle addition or subtraction formula.[Hint: This diagram of special trigonometry. values may help.].Choose answer:(A) (B) (C) (D)
Full solution
Q. Find exactly using an angle addition or subtraction formula.[Hint: This diagram of special trigonometry. values may help.].Choose answer:(A) (B) (C) (D)
- Express as Sum/Difference: We need to express as a sum or difference of angles for which we know the exact trigonometric values. The angles that are commonly used and for which we have exact values are , , and (and their multiples). We can write as the sum of and , which simplifies to .
- Apply Angle Addition Formula: Now we will use the angle addition formula for tangent, which is . Let and .
- Find Tangent Values: We need to find the exact values for and . From the unit circle, we know that . For , we know that is in the third quadrant where tangent is positive, and is the same as because they differ by an exact circle . Since , (because it's in the third quadrant where tangent is negative).
- Substitute Values: Now we substitute these values into the angle addition formula: $\tan\left(\frac{\(19\)\pi}{\(12\)}\right) = \tan\left(\frac{\(4\)\pi}{\(3\)} + \frac{\pi}{\(4\)}\right) = \frac{\tan\left(\frac{\(4\)\pi}{\(3\)}\right) + \tan\left(\frac{\pi}{\(4\)}\right)}{\(1\) - \tan\left(\frac{\(4\)\pi}{\(3\)}\right)\tan\left(\frac{\pi}{\(4\)}\right)} = \frac{(-\sqrt{\(3\)}) + \(1\)}{\(1\) - (-\sqrt{\(3\)})(\(1\))}.
- Simplify Expression: Simplify the expression: \(\tan\left(\frac{19\pi}{12}\right) = \frac{(-\sqrt{3}) + 1}{1 + \sqrt{3}}\).
- Rationalize Denominator: To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \((1 - \sqrt{3})\).
- Perform Multiplication: Perform the multiplication: \(\left((-\sqrt{3} + 1) \times (1 - \sqrt{3})\right) / \left((1 + \sqrt{3}) \times (1 - \sqrt{3})\right)\).
- Use Difference of Squares: Use the difference of squares formula for the denominator: \((1 - \sqrt{3})^2 = 1^2 - 2\cdot 1\cdot \sqrt{3} + (\sqrt{3})^2 = 1 - 2\sqrt{3} + 3\).
- Expand Numerator: Simplify the denominator: \(1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3}\).
- Final Simplification: Expand the numerator: \((-\sqrt{3} + 1) \times (1 - \sqrt{3}) = -\sqrt{3} + \sqrt{3}^2 - 1 + \sqrt{3} = -1 + 3 - \sqrt{3} = 2 - \sqrt{3}.\)
- Correct Division: Now we have: \(\tan\left(\frac{19\pi}{12}\right) = \frac{2 - \sqrt{3}}{4 - 2\sqrt{3}}\).
- Correct Division: Now we have: \(\tan\left(\frac{19\pi}{12}\right) = \frac{2 - \sqrt{3}}{4 - 2\sqrt{3}}\).To simplify the fraction, we divide both the numerator and the denominator by \(2\): \(\frac{2 - \sqrt{3}}{2} / \frac{4 - 2\sqrt{3}}{2} = \frac{1 - \sqrt{3}/2}{2 - \sqrt{3}}\).
- Correct Division: Now we have: \(\tan\left(\frac{19\pi}{12}\right) = \frac{2 - \sqrt{3}}{4 - 2\sqrt{3}}\).To simplify the fraction, we divide both the numerator and the denominator by \(2\): \(\frac{2 - \sqrt{3}}{2} / \frac{4 - 2\sqrt{3}}{2} = \frac{1 - \sqrt{3}/2}{2 - \sqrt{3}}\).We made a mistake in the previous step by dividing incorrectly. We should divide both the numerator and the denominator by \(2\) correctly: \(\frac{2 - \sqrt{3}}{2} / \frac{4 - 2\sqrt{3}}{2} = \frac{1 - \sqrt{3}/2}{2 - \sqrt{3}}\).
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