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Find cos((17 pi)/(12)) exactly using an angle addition or subtraction formula. [Hint: This diagram of special trigonometry. values may help.]. Choose 1 answer: (A) (sqrt2-sqrt6)/(4) (B) (-sqrt2-sqrt6)/(4) (C) (-sqrt2+sqrt6)/(4) (D) (sqrt2+sqrt6)/(4)

Find cos⁡(17π12) \cos \left(\frac{17 \pi}{12}\right) exactly using an angle addition or subtraction formula.\newline[Hint: This diagram of special trigonometry. values may help.].\newlineChoose 11 answer:\newline(A) 2−64 \frac{\sqrt{2}-\sqrt{6}}{4} \newline(B) −2−64 \frac{-\sqrt{2}-\sqrt{6}}{4} \newline(C) −2+64 \frac{-\sqrt{2}+\sqrt{6}}{4} \newline(D) 2+64 \frac{\sqrt{2}+\sqrt{6}}{4}

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Q. Find cos⁡(17π12) \cos \left(\frac{17 \pi}{12}\right) exactly using an angle addition or subtraction formula.\newline[Hint: This diagram of special trigonometry. values may help.].\newlineChoose 11 answer:\newline(A) 2−64 \frac{\sqrt{2}-\sqrt{6}}{4} \newline(B) −2−64 \frac{-\sqrt{2}-\sqrt{6}}{4} \newline(C) −2+64 \frac{-\sqrt{2}+\sqrt{6}}{4} \newline(D) 2+64 \frac{\sqrt{2}+\sqrt{6}}{4}
  1. Expressing (17π12)(\frac{17\pi}{12}) as a sum or difference: We need to express (17π12)(\frac{17\pi}{12}) as a sum or difference of angles for which we know the cosine values exactly. The angles we can use are typically multiples of π6\frac{\pi}{6}, π4\frac{\pi}{4}, or π3\frac{\pi}{3} because these correspond to special angles in the unit circle (30∘30^\circ, 45∘45^\circ, and 60∘60^\circ respectively).
  2. Using the angle addition formula for cosine: We can write (17π12)(\frac{17\pi}{12}) as the sum of (4π12)(\frac{4\pi}{12}) and (13π12)(\frac{13\pi}{12}). Since (4π12)(\frac{4\pi}{12}) simplifies to (π3)(\frac{\pi}{3}) and (13π12)(\frac{13\pi}{12}) simplifies to (π4+π6)(\frac{\pi}{4} + \frac{\pi}{6}), we can use the angle addition formula for cosine to find the exact value of cos⁡(17π12)\cos(\frac{17\pi}{12}).
  3. Finding the cosine and sine values: The angle addition formula for cosine is cos⁡(α+β)=cos⁡(α)cos⁡(β)−sin⁡(α)sin⁡(β)\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta). We will use α=π3\alpha = \frac{\pi}{3} and β=π4+π6\beta = \frac{\pi}{4} + \frac{\pi}{6}.
  4. Finding the cosine and sine of (π/4+π/6)(\pi/4 + \pi/6): First, we need to find the cosine and sine values for π/3\pi/3, π/4\pi/4, and π/6\pi/6. From the unit circle, we know that:\newlinecos⁡(π/3)=1/2,\cos(\pi/3) = 1/2,\newlinesin⁡(π/3)=3/2,\sin(\pi/3) = \sqrt{3}/2,\newlinecos⁡(π/4)=2/2,\cos(\pi/4) = \sqrt{2}/2,\newlinesin⁡(π/4)=2/2,\sin(\pi/4) = \sqrt{2}/2,\newlinecos⁡(π/6)=3/2,\cos(\pi/6) = \sqrt{3}/2,\newlinesin⁡(π/6)=1/2.\sin(\pi/6) = 1/2.
  5. Calculating cos⁡(17π12)\cos\left(\frac{17\pi}{12}\right) using the angle addition formula: Now we need to find the cosine and sine of (π4+π6)\left(\frac{\pi}{4} + \frac{\pi}{6}\right) using the angle addition formulas for cosine and sine:\newlinecos⁡(π4+π6)=cos⁡(π4)cos⁡(π6)−sin⁡(π4)sin⁡(π6),\cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right),\newlinesin⁡(π4+π6)=sin⁡(π4)cos⁡(π6)+cos⁡(π4)sin⁡(π6).\sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right).
  6. Substituting the known values into the formula: Substitute the known values into the formulas:\newlinecos⁡(π4+π6)=(22)(32)−(22)(12)=(64)−(24)\cos(\frac{\pi}{4} + \frac{\pi}{6}) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2}) = (\frac{\sqrt{6}}{4}) - (\frac{\sqrt{2}}{4}),\newlinesin⁡(π4+π6)=(22)(32)+(22)(12)=(64)+(24)\sin(\frac{\pi}{4} + \frac{\pi}{6}) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = (\frac{\sqrt{6}}{4}) + (\frac{\sqrt{2}}{4}).
  7. Performing the multiplication: Now we can find cos⁡(17π12)\cos\left(\frac{17\pi}{12}\right) using the angle addition formula with α=π3\alpha = \frac{\pi}{3} and β=π4+π6\beta = \frac{\pi}{4} + \frac{\pi}{6}:cos⁡(17π12)=cos⁡(π3)cos⁡(π4+π6)−sin⁡(π3)sin⁡(π4+π6)\cos\left(\frac{17\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4} + \frac{\pi}{6}\right) - \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right).
  8. Simplifying the expression: Substitute the known values into the formula: \cos\left(\frac{\(17\)\pi}{\(12\)}\right) = \left(\frac{\(1\)}{\(2\)}\right)\left(\frac{\sqrt{\(6\)}}{\(4\)} - \frac{\sqrt{\(2\)}}{\(4\)}\right) - \left(\frac{\sqrt{\(3\)}}{\(2\)}\right)\left(\frac{\sqrt{\(6\)}}{\(4\)} + \frac{\sqrt{\(2\)}}{\(4\)}\right).
  9. Simplifying the expression: Substitute the known values into the formula: \(\newlinecos⁡(17π12)=(12)(64−24)−(32)(64+24)\cos\left(\frac{17\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\right).Perform the multiplication: \newline\cos\left(\frac{17\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{6}}{4}\right) - \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{6}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{4}\right),\(\newline\\cos\left(\frac{17\pi}{12}\right) = \frac{\sqrt{6}}{8} - \frac{\sqrt{2}}{8} - \frac{\sqrt{18}}{8} - \frac{\sqrt{6}}{8}\).
  10. Simplifying the expression: Substitute the known values into the formula: \(\cos\left(\frac{17\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\right)\). Perform the multiplication: \(\cos\left(\frac{17\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{6}}{4}\right) - \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{6}}{4}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{4}\right), \)\cos\left(\frac{\(17\)\pi}{\(12\)}\right) = \frac{\sqrt{\(6\)}}{\(8\)} - \frac{\sqrt{\(2\)}}{\(8\)} - \frac{\sqrt{\(18\)}}{\(8\)} - \frac{\sqrt{\(6\)}}{\(8\)}\(. Simplify the expression: \)\cos\left(\frac{\(17\)\pi}{\(12\)}\right) = \frac{\sqrt{\(6\)}}{\(8\)} - \frac{\sqrt{\(2\)}}{\(8\)} - \frac{\(3\)\sqrt{\(2\)}}{\(8\)} - \frac{\sqrt{\(6\)}}{\(8\)}, \(\cos\left(\frac{17\pi}{12}\right) = - \frac{\sqrt{2}}{8} - \frac{3\sqrt{2}}{8}, \)\cos\left(\frac{\(17\)\pi}{\(12\)}\right) = - \frac{\(4\)\sqrt{\(2\)}}{\(8\)}, \cos\left(\frac{1717\pi}{1212}\right) = - \frac{\sqrt{22}}{22}.

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