Problem 13 - Contemporary Precalculus

Question

Rewrite the given expression in terms of \(\sin x\) and \(\cos x\). $$\sin \left(\frac{\pi}{2}+x\right)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\sin \left(\frac{\pi}{2}+x\right)\) in terms of \(\sin x\) and \(\cos x\). Answer: \(\sin\left(\frac{\pi}{2}+x\right) = \cos(x)\)

Step 1: Identify A and B

In our given expression, \(\sin \left(\frac{\pi}{2}+x\right)\), we have: \(A = \frac{\pi}{2}\) and \(B = x\)

Step 2: Use the sum of angles formula for sine

Now, we will apply the sum of angles formula for sine to our given expression, \(\sin \left(\frac{\pi}{2}+x\right)\): $$\sin\left(\frac{\pi}{2}+x\right) = \sin\left(\frac{\pi}{2}\right)\cos(x) + \cos\left(\frac{\pi}{2}\right)\sin(x)$$

Step 3: Simplify the expression

Since \(\sin\left(\frac{\pi}{2}\right) = 1\) and \(\cos\left(\frac{\pi}{2}\right) = 0\), we can substitute these values into our expression: $$\sin\left(\frac{\pi}{2}+x\right) = (1)\cos(x) + (0)\sin(x)$$

Step 4: Write the final expression

Our final expression in terms of \(\sin x\) and \(\cos x\) is: $$\sin\left(\frac{\pi}{2}+x\right) = \cos(x)$$

Key Concepts

Sum of Angles Formula

The Sum of Angles Formula is a fundamental part of trigonometry. It allows us to express trigonometric functions of the sum of two angles in terms of the functions of the individual angles. Specifically, for the sine function, this formula is written as:

• \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)

Here, the angles \(A\) and \(B\) are added together, and this formula helps to break the sine of this sum into the products of sine and cosine of the original angles.
This formula is valuable for simplifying expressions and solving various trigonometric equations. It often finds use in signal processing, engineering, and physics as well.
Applying this formula to different trigonometric problems helps in better understanding and solving complex angles and expressions.

Sine Function

The sine function, represented as \( \sin(\theta) \), is one of the basic trigonometric functions. It describes the vertical coordinate of a point on the unit circle.
In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

The sine function displays periodic behavior, repeating its values in a smooth wave-like pattern every 360 degrees or \( 2\pi \) radians.
It's important in modeling oscillatory motions like waves and is essential in fields such as astronomy and navigation. Understanding its behavior and transformations, like shifts and amplitude changes, is key to manipulating trigonometric expressions.

Cosine Function

The cosine function, denoted as \( \cos(\theta) \), is another primary trigonometric function. It represents the horizontal coordinate of a point on the unit circle.
For a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse:

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

Cosine functions also exhibit periodic behavior, identical to the sine function, with a period of 360 degrees or \( 2\pi \) radians.
The cosine wave is crucial in describing wave motion and is widely used in physics, engineering, and computer graphics. By analyzing and manipulating cosine properties, complex systems and rotations can be easily explored and understood.

Angle Addition Formula

The Angle Addition Formula is a powerful tool used to expand trigonometric functions involving sums and differences of angles. It provides an explicit way to calculate trigonometric values for two combined angles:

• The Sine Addition Formula is: \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)
• The Cosine Addition Formula is: \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)

These formulas are not only useful in solving equations but also in deriving other identities and simplifying expressions.
For example, they are vital in deriving double angle and half-angle formulas. Understanding these core principles is essential for anyone studying trigonometry or dealing with periodic and oscillatory systems.