Problem 133 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. CONCEPT CHECK True or false? For all angles \(\theta\), \(\sin \theta+\cos \theta=1 .\) If false, give an example showing why.

Step-by-Step Solution

Answer

False; for \( \theta = 45^\circ \), \( \sin \theta + \cos \theta = \sqrt{2} \).

Step 1: Understand the Trigonometric Identity

The statement in the problem checks if the identity \( \sin \theta + \cos \theta = 1 \) holds for all angles \( \theta \). This is different from the known identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which is always true.

Step 2: Calculate Using a Specific Angle

Let's test the statement with a specific angle, such as \( \theta = 45^\circ \). For \( \theta = 45^\circ \), we know that \( \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \). Thus, \( \sin 45^\circ + \cos 45^\circ = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are essential in the study of waves, oscillations, and many other applications in science and engineering. The primary trigonometric functions are sine (\(\sin\theta\)) and cosine (\(\cos\theta\)), which are based on a right-angled triangle. The sine function represents the ratio of the length of the side opposite to the angle to the hypotenuse, while the cosine function relates the adjacent side to the hypotenuse.
These functions are periodic, meaning they repeat their values in regular intervals. For example, both sine and cosine functions have a period of \(360^\circ\) or \(2\pi\) radians. This periodic nature is helpful when dealing with circular and oscillatory motion. Additionally, these functions are foundational to trigonometric identities, which are equations that are true for all values within their defined domains. Understanding these identities is crucial for solving a wide range of mathematical problems.
Some common trigonometric identities include the Pythagorean identity (\(\sin^2\theta + \cos^2\theta = 1\)) and the angle sum identities, which allow for the calculation of sine and cosine for the sum of two angles. These identities also assist in simplifying complex trigonometric expressions.

Angle Calculation

Angle calculation in trigonometry involves determining the measure of an angle based on known values using trigonometric functions. Whether you're solving for an angle in a right triangle or along a unit circle, understanding how to calculate angles is fundamental in trigonometry. When calculating angles, it is useful to remember some key angle values such as \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\), as well as their corresponding radian values.
To illustrate, if you want to verify the statement \(\sin\theta + \cos\theta = 1\), using a specific angle can help. For \(\theta = 45^\circ\), \(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\). Thus, adding these values gives \(\sqrt{2}\), which is not equal to 1, showing a discrepancy from the given assertion. This calculation reinforces the importance of using angle-specific values to evaluate mathematical statements accurately.
Additionally, angle calculations can involve the use of inverse trigonometric functions when determining unknown angles from trigonometric ratios. This is particularly useful in real-life applications such as engineering, where precise angle measurements are vital.

Mathematical Proof

Mathematical proof is a logical process used to demonstrate the truth or falsehood of a given mathematical statement. Proofs are essential in verifying mathematical theorems and identities, ensuring that they hold under predefined conditions. They enhance the understanding and reliability of mathematical concepts.
When using a mathematical proof to evaluate a trigonometric statement like \(\sin\theta + \cos\theta = 1\), it involves checking the truth of the statement against known identities. For instance, using a specific angle in angle calculation can serve as a quick method to disprove an incorrect identity like this one. Through substitution and calculation, if the resulting value does not match the desired outcome, it indicates the original statement is false.
In our example, substituting \(\theta = 45^\circ\) leads to \(\sin 45^\circ + \cos 45^\circ = \sqrt{2}\), proving that the statement does not hold true for this angle, and in general, does not hold for all angles. Mathematical proofs, such as this angle verification, are tools that provide clarity and confirm the integrity of mathematical assertions. In learning contexts, proving false statements helps learners identify and appreciate correct identities.