Evaluate each expression without using a calculator.

tan (arccos 3/4)

검증된 단계별 안내

Recognize that the expression is \( \tan(\arccos(\frac{3}{4})) \). Let \( \theta = \arccos(\frac{3}{4}) \), which means \( \cos \theta = \frac{3}{4} \) and \( \theta \) is an angle in the range \([0, \pi]\).

Use the Pythagorean identity to find \( \sin \theta \). Since \( \sin^2 \theta + \cos^2 \theta = 1 \), substitute \( \cos \theta = \frac{3}{4} \) to get \( \sin^2 \theta = 1 - \left(\frac{3}{4}\right)^2 \).

Calculate \( \sin \theta = \sqrt{1 - \left(\frac{3}{4}\right)^2} \). Since \( \theta \) is in the range \([0, \pi]\) and \( \arccos \) returns values in the first or second quadrant, \( \sin \theta \) is positive.

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the values of \( \sin \theta \) and \( \cos \theta \) to express \( \tan(\arccos(\frac{3}{4})) \) as a fraction.

Simplify the fraction to get the exact value of \( \tan(\arccos(\frac{3}{4})) \) without using a calculator.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccos, return the angle whose trigonometric ratio is given. For example, arccos(3/4) gives the angle whose cosine is 3/4. Understanding this allows you to translate the problem into finding trigonometric values of specific angles.

Right Triangle Relationships

Using the value of cosine as the ratio of adjacent side over hypotenuse, you can construct a right triangle to find the other sides. This helps in determining sine and tangent values by applying the Pythagorean theorem to find the missing side lengths.

Definition of Tangent in Terms of Sine and Cosine

Tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Once sine and cosine values are known or derived, you can compute tangent without a calculator by simple fraction operations.

Evaluate each expression without using a calculator.tan (arccos 3/4)