Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of the expression without using a calculator. $$\cos \left(\sin ^{-1} 1+\cos ^{-1} 0\right)$$

Step-by-Step Solution

Answer

-1

Step 1: Identify Inverse Trigonometric Values

Firstly, identify the inverse trigonometric values. Here, \(\sin^{-1}(1) = \frac{\pi}{2}\) and \(\cos^{-1}(0) = \frac{\pi}{2}\). These known inverse trigonometric values will be used for further calculations.

Step 2: Calculate the angle

Next calculate the angle by adding the angles resulted from inverse trigonometric values. Therefore, \(\frac{\pi}{2} + \frac{\pi}{2} = \pi\). So, the argument inside the \(\cos\) function is \(\pi\).

Step 3: Find the Value of the Expression

Finally, calculate the cosine of \(\pi\), i.e. \(\cos(\pi)\). The result is \(-1\), because the cosine of \(\pi\) equals \(-1\).

Key Concepts

sine inverse

The sine inverse, also known as arcsine, is a function that helps us determine an angle whose sine is a given number. In mathematical terms, this can be expressed as follows:

• If \( y = \sin^{-1}(x) \), then \( \sin(y) = x \).

This function is only defined within certain limits due to the nature of how sine and arcsine functions work. Typically, the sine inverse is restricted to

• \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\).

In the exercise provided, we calculated \(\sin^{-1}(1)\). This gives us \(\frac{\pi}{2}\), because \(\sin\left(\frac{\pi}{2}\right) = 1\). Knowing the common values of arcsine can be quite useful for solving trigonometric identities and equations without the need for complex calculations or calculators.

cosine inverse

The cosine inverse, also known as arccosine, is a function that allows us to find an angle whose cosine is a given value. Mathematically, we express this as:

• If \( y = \cos^{-1}(x) \), then \( \cos(y) = x \).

Similar to sine inverse, this function also has certain restrictions due to the properties of cosine and arccosine.

• The cosine inverse is typically defined between \(0 \le y \le \pi\).

In the exercise, the value \(\cos^{-1}(0)\) was calculated, which equals \(\frac{\pi}{2}\), because \(\cos\left(\frac{\pi}{2}\right) = 0\). Being familiar with these standard values makes solving expressions quicker and enables us to evaluate trigonometric expressions without needing a calculator.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variables involved. These identities are useful in simplifying and solving equations involving trigonometric functions.One of the most common identities is the Pythagorean identity:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

This identity is crucial in understanding relationships between sine and cosine and can be used to derive other useful identities.In our case, knowing that the cosine of \(\pi\) is \(-1\) helped us evaluate the expression \(\cos\left(\sin^{-1}(1) + \cos^{-1}(0)\right)\). By understanding that \(\sin^{-1}(1)\) and \(\cos^{-1}(0)\) both equal \(\frac{\pi}{2}\), we added these angles to get \(\pi\), for which the cosine is well-known to be \(-1\). Mastery of these identities enables efficient and effective problem-solving in trigonometry.