find exact value using half-angle formulas tan(-3pi/8)
The Correct Answer and Explanation is:
To find the exact value of tan(−3π/8) using half-angle formulas, we begin by recognizing that:tan(x2)=sinx1+cosxortan(x2)=1−cosxsinx\tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x} \quad \text{or} \quad \tan\left(\frac{x}{2}\right) = \frac{1 – \cos x}{\sin x}tan(2x)=1+cosxsinxortan(2x)=sinx1−cosx
We want to express tan(−3π/8) using a half-angle identity. Note that:−3π/8=12(−3π/4)−3\pi/8 = \frac{1}{2}(−3\pi/4)−3π/8=21(−3π/4)
So, we can write:tan(−3π8)=tan(12⋅(−3π4))\tan\left(-\frac{3\pi}{8}\right) = \tan\left(\frac{1}{2} \cdot \left(-\frac{3\pi}{4}\right)\right)tan(−83π)=tan(21⋅(−43π))
Now let us use the identity:tan(x2)=sinx1+cosx\tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}tan(2x)=1+cosxsinx
Let x = −3π/4. Then:
- sin(−3π/4) = −sin(3π/4) = −√2/2
- cos(−3π/4) = cos(3π/4) = −√2/2
So we substitute:tan(−3π8)=sin(−3π/4)1+cos(−3π/4)=−221+(−22)\tan\left(-\frac{3\pi}{8}\right) = \frac{\sin(-3\pi/4)}{1 + \cos(-3\pi/4)} = \frac{-\frac{\sqrt{2}}{2}}{1 + (-\frac{\sqrt{2}}{2})}tan(−83π)=1+cos(−3π/4)sin(−3π/4)=1+(−22)−22
Now simplify the denominator:1−22=2−221 – \frac{\sqrt{2}}{2} = \frac{2 – \sqrt{2}}{2}1−22=22−2
So the full expression becomes:tan(−3π8)=−222−22=−2/2(2−2)/2\tan\left(-\frac{3\pi}{8}\right) = \frac{-\frac{\sqrt{2}}{2}}{\frac{2 – \sqrt{2}}{2}} = \frac{-\sqrt{2}/2}{(2 – \sqrt{2})/2}tan(−83π)=22−2−22=(2−2)/2−2/2
Now cancel the denominators:tan(−3π8)=−22−2\tan\left(-\frac{3\pi}{8}\right) = \frac{-\sqrt{2}}{2 – \sqrt{2}}tan(−83π)=2−2−2
To simplify, multiply numerator and denominator by the conjugate of the denominator:−22−2⋅2+22+2=−2(2+2)(2−2)(2+2)\frac{-\sqrt{2}}{2 – \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{-\sqrt{2}(2 + \sqrt{2})}{(2 – \sqrt{2})(2 + \sqrt{2})}2−2−2⋅2+22+2=(2−2)(2+2)−2(2+2)
Simplify numerator:−2(2+2)=−22−2-\sqrt{2}(2 + \sqrt{2}) = -2\sqrt{2} – 2−2(2+2)=−22−2
Simplify denominator:(2−2)(2+2)=4−2=2(2 – \sqrt{2})(2 + \sqrt{2}) = 4 – 2 = 2(2−2)(2+2)=4−2=2
So:tan(−3π8)=−22−22=−2−1\tan\left(-\frac{3\pi}{8}\right) = \frac{-2\sqrt{2} – 2}{2} = -\sqrt{2} – 1tan(−83π)=2−22−2=−2−1
Final Answer:tan(−3π8)=−2−1\tan\left(-\frac{3\pi}{8}\right) = -\sqrt{2} – 1tan(−83π)=−2−1
Explanation (300 words):
To find the exact value of tan(−3π/8), we apply the half-angle identity. This identity allows us to express the tangent of half an angle in terms of sine and cosine of the full angle. Recognizing that −3π/8 is half of −3π/4 gives us a good starting point. This means we can let x = −3π/4 and use the identity:tan(x2)=sinx1+cosx\tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x}tan(2x)=1+cosxsinx
We find the sine and cosine of −3π/4. Using unit circle knowledge, sin(−3π/4) equals −√2/2 and cos(−3π/4) equals −√2/2. Substituting into the formula, we simplify the expression step by step. After eliminating denominators and rationalizing, we arrive at:tan(−3π/8)=−2−1\tan(-3\pi/8) = -\sqrt{2} – 1tan(−3π/8)=−2−1
This is the exact value without any decimal approximation. The negative sign reflects the fact that the angle lies in the fourth quadrant, where tangent values are negative. This method avoids the use of calculators and relies solely on algebraic identities and known trigonometric values from the unit circle.
