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Math  /  Trigonometry

QuestionUse a compound angle formula to determine the exact value of tan⁡π12\tan \frac{\pi}{12}

Studdy Solution

STEP 1

What is this asking? We need to find the *exact* value of tan⁡π12\tan \frac{\pi}{12} using a *compound angle formula*. Watch out! Remember, π12\frac{\pi}{12} is not one of our special angles, but we can rewrite it as a sum or difference of angles we *do* know!
Also, "exact" means no decimals!

STEP 2

1. Rewrite the Angle
2. Apply the Tangent Compound Angle Formula
3. Simplify Using Known Values
4. Rationalize the Denominator

STEP 3

We want to express π12\frac{\pi}{12} using special angles.
We know the tangent values for angles like π6\frac{\pi}{6}, π4\frac{\pi}{4}, and π3\frac{\pi}{3}.
Let's think about how we can make π12\frac{\pi}{12} using these!

STEP 4

We can write π12\frac{\pi}{12} as a difference: π12=π3−π4\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} because 4π12−3π12=π12\frac{4\pi}{12} - \frac{3\pi}{12} = \frac{\pi}{12}.
Awesome!

STEP 5

The tangent compound angle formula for difference of angles is: tan⁡(a−b)=tan⁡a−tan⁡b1+tan⁡a⋅tan⁡b \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b} Let's use it with a=π3a = \frac{\pi}{3} and b=π4b = \frac{\pi}{4}.

STEP 6

Substituting our values, we get: tan⁡(π3−π4)=tan⁡π3−tan⁡π41+tan⁡π3⋅tan⁡π4 \tan\left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \frac{\tan\frac{\pi}{3} - \tan\frac{\pi}{4}}{1 + \tan\frac{\pi}{3} \cdot \tan\frac{\pi}{4}}

STEP 7

We know that tan⁡π3=3\tan\frac{\pi}{3} = \sqrt{3} and tan⁡π4=1\tan\frac{\pi}{4} = 1.
Let's plug these in!

STEP 8

tan⁡π12=3−11+3⋅1=3−11+3 \tan\frac{\pi}{12} = \frac{\sqrt{3} - 1}{1 + \sqrt{3} \cdot 1} = \frac{\sqrt{3} - 1}{1 + \sqrt{3}}

STEP 9

To get rid of the square root in the denominator, we'll multiply the top and bottom by the conjugate of the denominator, which is 1−31 - \sqrt{3}.

STEP 10

3−11+3⋅1−31−3=3−3−1+31−3=23−4−2 \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{\sqrt{3} - 3 - 1 + \sqrt{3}}{1 - 3} = \frac{2\sqrt{3} - 4}{-2}

STEP 11

Dividing both terms in the numerator by −2-2, we get: 23−2−4−2=−3+2=2−3 \frac{2\sqrt{3}}{-2} - \frac{4}{-2} = -\sqrt{3} + 2 = 2 - \sqrt{3}

STEP 12

The exact value of tan⁡π12\tan\frac{\pi}{12} is 2−32 - \sqrt{3}.

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