Problem 100 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \sin 45^{\circ}+\cos 45^{\circ}=1 $$

Step-by-Step Solution

Answer

The given statement is false. The corrected statement is: \(\sin 45^{\circ} + \cos 45^{\circ} = \sqrt{2}\).

Step 1: Identify the values of trigonometric functions

The values of \(\sin 45^{\circ}\) and \(\cos 45^{\circ}\) are both \(\frac{\sqrt{2}}{2}\).

Step 2: Substitute identified values

Now substitute these values into the given statement. We obtain: \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 1\).

Step 3: Simplify the expression

On simplifying we get \(\sqrt{2} \neq 1\). Thus, the original statement is false.

Step 4: Correct the statement

Clearly, the correct statement should be: \(\sin 45^{\circ} + \cos 45^{\circ} = \sqrt{2}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, particularly in geometry and physics. They relate the angles of a triangle to the ratios of its sides. The most commonly known trigonometric functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)). Each of these functions can be understood as a relationship within a right-angled triangle.

• Sine (\(sin\theta\)) measures the ratio of the length of the side opposite the angle to the hypotenuse.
• Cosine (\(cos\theta\)) measures the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(tan\theta\)) is the ratio of the sine to the cosine of the angle.

Knowing these functions helps you solve triangles, calculate lengths, and understand waves, among many other applications. They are the foundation for more advanced mathematical concepts.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and are solved for angles. They can be straightforward, like finding basic angles, or more complex, involving multiple functions and identities. Solving these equations often requires knowledge of standard angles and identities.

• Example: Equation involving sin or cos like \(sin\theta = \frac{1}{2}\)
• You solve by finding all angles that satisfy the equation, often within one rotation (0 to 360 degrees).

Trigonometric identities, such as Pythagorean identities, can simplify equations and help find solutions. These identities express relations between different trigonometric functions. For instance, the identity \(\sin^2\theta + \cos^2\theta = 1\) helps manipulate expressions to find more angles or verify solutions.

Sin and Cos Values

Specific angles have well-known sine and cosine values, often referred to as special angles. Memorizing these angles can make solving equations and understanding identities easier. For example:

• \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)

In the problem provided, these values are used to test the truth of a mathematical statement. The combination of \(\sin 45^{\circ}\) and \(\cos 45^{\circ}\) results in \(\sqrt{2}\), demonstrating the importance of knowing these values. This helps in verifying and correcting equations, showcasing the practicality of keeping these numbers handy when working with angles and trigonometric identities.