Evaluate each expression without using a calculator.

tan (arccos 3/4)

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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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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.

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Introduction to Inverse Trig Functions

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.

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30-60-90 Triangles

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.

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