Question

Without using a calculator, evaluate, if possible, the following expressions. $$\cos \left(\cos ^{-1}(-1)\right)$$

Short answer

Question: Evaluate the expression \(\cos \left(\cos ^{-1}(-1)\right)\) without using a calculator. Answer: -1

Step-by-step solution

Understand the inverse cosine function

The inverse cosine function, \(\cos^{-1}(x)\), returns the angle, call it \(\theta\), whose cosine is \(x\). The function domain is \(-1 \leq x \leq 1\), and the resulting angle range is \(0 \leq \theta \leq \pi\). We need to find the angle for which \(\cos \theta = -1\).

Find the angle whose cosine is -1

Recall that the cosine function is negative in the second and third quadrants. Since the output of the inverse cosine function is between \(0\) and \(\pi\), we only need to consider the second quadrant. The angle for which \(\cos \theta = -1\) in the second quadrant is \(\theta = \pi\). Thus, \(\cos ^{-1}(-1) = \pi\).

Find the cosine of the angle found in step 2

Now that we have the angle, we need to evaluate the cosine of that angle. We have \(\cos \left(\cos ^{-1}(-1)\right) = \cos \pi\). We know that \(\cos \pi = -1\).

Determine the final result of the expression

Finally, we have found that \(\cos \left(\cos ^{-1}(-1)\right) = \cos \pi = -1\). So, without using a calculator, the expression evaluates to \(-1\).

Key concepts

Understanding the Cosine Function

The Cosine function (often denoted as \( \cos \)) is one of the six fundamental trigonometric functions. It is crucial in understanding relationships between angles and sides in right-angled triangles.
In a triangle, the cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle. This can be written as:

• \( \cos \theta = \frac{adjacent}{hypotenuse} \)

The range of the cosine function is from -1 to 1. This means that the output of \( \cos \theta \) will always lie within this interval.
Cosine has a period of \( 2\pi \), which means that it repeats its values every \( 2\pi \) radians (or 360 degrees). This is key for solving trigonometric equations over different intervals. Recognizing this periodic behavior helps you understand how angles above \( 360^\circ \) or below \( 0^\circ \) can make the same cosine values.
Cosine is particularly handy because it's even, meaning \( \cos(-\theta) = \cos(\theta) \). This property simplifies many calculations, especially when dealing with symmetrical relationships in problems.

Exploring Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value within their domains. Understanding these identities can simplify complex trigonometry problems.

• The Pythagorean Identity, one of the most essential, states: \( \cos^2 \theta + \sin^2 \theta = 1 \).
• Another useful identity involves the cosine function and is the Complementary Angle Identity: \( \cos(90^\circ - \theta) = \sin \theta \).
• A powerful aspect of cosine lies in its relation with the other trigonometric functions through the angle sum identities such as: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Mastering these identities allows you to transform and simplify trigonometric expressions. It also helps in solving equations and proving other mathematical properties. Recognizing the equivalences provided by identities supports a deeper understanding of how different angles relate through their trigonometric function values.

Navigating Angle Measurement

Angle measurement is a fundamental element in trigonometry and geometry. Angles can be measured in different units, most commonly in degrees and radians.
Degrees measure angles based on dividing a circle into 360 parts. This is often intuitive because of its widespread usage in everyday contexts, such as in clocks.
Radians measure angles based on the radius of a circle and are the standard unit of angular measure in mathematics. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius. In terms of conversion, \( 180^\circ \) equals \( \pi \) radians.

• When calculating angles, especially for trigonometric functions, using radians simplifies many formulas, because radians are directly tied to the unit circle.

Understanding these measurements and being able to convert between them is critical when dealing with problems involving trigonometry, such as finding the cosine of an angle. This ability to transition between these measurements provides flexibility and precision in solving mathematical problems accurately.