Problem 58 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of each expression. $$ \tan \left[\cos ^{-1}\left(-\frac{1}{4}\right)\right] $$

Step-by-Step Solution

Answer

-\sqrt{15}

Step 1: Find angle from inverse Cosine function

To solve \( \tan (\cos^{-1} (-\frac{1}{4})) \), first find the value of \( \cos^{-1} (-\frac{1}{4}) \), which is the angle whose cosine is -1/4. Let this angle be \( \theta \). So, \( \theta = \cos^{-1} (-\frac{1}{4}) \).

Step 2: Sketch a triangle

Now, represent this in a triangle. Since cosine is negative, we're in the second quadrant where \( x < 0 \) and \( y > 0 \). Draw a right triangle in the second quadrant with \( \theta \) as the reference angle, the adjacent side as -1 (due to negative cosine value), and the hypotenuse as 4.

Step 3: Find the third side (opposite) of the triangle using Pythagorean Theorem

The third side of the triangle can be determined using the Pythagorean theorem. Let the opposite side be \( o \). So according to the Pythagorean theorem, \( o = \sqrt{hypotenuse^2 - base^2} = \sqrt{4^2 - (-1)^2} = \sqrt{15} \).

Step 4: Find the tangent of the angle

Now in the triangle, \( \tan(\theta) \) is defined as 'opposite over adjacent', i.e., \( o/-1 = -o = -\sqrt{15} \). So, \( \tan(\cos^{-1}(-1/4)) = -\sqrt{15} \).

Key Concepts

Tangent of an Angle

The tangent of an angle, often abbreviated as 'tan', is one of the fundamental trigonometric ratios in right triangle trigonometry. It is defined as the ratio of the side opposite to the angle in question to the side adjacent to that angle. For any angle \( \theta \), the tangent can be represented as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Understanding the tangent is not just about memorizing this ratio, but also knowing how to use it in various quadrants of the coordinate plane. In the context of an exercise where we're given an inverse cosine, we may need to sketch a triangle to help visualize the problem and determine the signs of the sides.

Pythagorean Theorem

The Pythagorean theorem is a critical concept in mathematics, especially used in geometry, to find the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite to the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem can be written as:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. When you are asked to find an unknown side in a right triangle, you will often use the Pythagorean theorem, and this principle is essential for solving many trigonometric problems.

Angle from Inverse Cosine

Finding the angle from an inverse cosine—denoted as \( \cos^{-1} \)—involves figuring out the angle whose cosine is a given value. This function is also called arccosine. Whenever you encounter the expression \( \cos^{-1}(x) \), you're essentially looking for an angle \( \theta \) such that \( \cos(\theta) = x \).

It's also important to note that the range of the inverse cosine function is from \(0\) to \(\pi\) radians, which means it provides angles from the first and second quadrants. Therefore, if the cosine value is negative, as in our exercise, the resulting angle will be in the second quadrant, where sine values are positive and cosine values are negative.

Right Triangle Trigonometry

Right triangle trigonometry is the study of the relationships between the angles and sides of right triangles. The main trigonometric functions involved are sine, cosine, and tangent, which are based on an angle of the triangle. Through these functions, you can solve for unknown sides or angles, given some known values. This branch of trigonometry is essential for solving many geometric problems.

For example, using the given information like the inverse cosine value in the exercise, one can not only find the angle but also determine the lengths of the sides by constructing a right triangle and applying the trigonometric definitions and the Pythagorean theorem, as demonstrated in the solution steps. This integration of various concepts allows for a comprehensive problem-solving approach in trigonometry.