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Chapter 5: Problem 84

Show that $$ \sin \left(t+\frac{\pi}{2}\right)=\cos t $$ for every number \(t\).

Short Answer

Expert verified
Using the angle sum formula for sine, \(\sin(A + B) = \sin A \cos B + \cos A \sin B\), we find the value of \(\sin\left(t + \frac{\pi}{2}\right)\) as follows: \[\sin\left(t + \frac{\pi}{2}\right) = \sin t \cos \frac{\pi}{2} + \cos t \sin \frac{\pi}{2}\] Since \(\sin \frac{\pi}{2} = 1\) and \(\cos \frac{\pi}{2} = 0\), we plug these values into the expression and simplify: \[\sin\left(t + \frac{\pi}{2}\right) = \sin t \cdot 0 + \cos t \cdot 1 = \cos t\] Thus, we show that \(\sin\left(t + \frac{\pi}{2}\right) = \cos t\) for every number \(t\).

Step by step solution

01

Write down the angle sum formula for sine

The angle sum formula for sine is given by: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] Now we will use this formula to find the value of \(\sin\left(t + \frac{\pi}{2}\right)\).
02

Rewrite the given equation using the angle sum formula

We want to find: \[ \sin\left(t + \frac{\pi}{2}\right) \] Using the angle sum formula, we replace \(A\) with \(t\) and \(B\) with \(\frac{\pi}{2}\). Thus, we get: \[ \sin\left(t + \frac{\pi}{2}\right) = \sin t \cos \frac{\pi}{2} + \cos t \sin \frac{\pi}{2} \]
03

Use the values of \(\sin \frac{\pi}{2}\) and \(\cos \frac{\pi}{2}\)

By evaluating the trigonometric values at these specific angles, we have: \[ \sin \frac{\pi}{2} = 1 \quad \text{and} \quad \cos \frac{\pi}{2} = 0 \] Now, we plug these values into the expression from Step 2: \[ \sin\left(t + \frac{\pi}{2}\right) = \sin t \cdot 0 + \cos t \cdot 1 \]
04

Simplify the expression and obtain the result

Since \(\sin t \cdot 0 = 0\), the expression simplifies to: \[ \sin\left(t + \frac{\pi}{2}\right) = \cos t \] So, we have shown that \(\sin\left(t + \frac{\pi}{2}\right) = \cos t\) for every number \(t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Sum Formula
The angle sum formula is a critical tool in trigonometry for dealing with expressions involving the sum of angles. Particularly for sine, the formula is expressed as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). This formula helps us break down the sine of a combined angle into components that we can more easily evaluate. It involves multiplying the sine and cosine components of each angle, \(A\) and \(B\), in a specific combination.
Understanding this formula is crucial because it allows us to decompose complex expressions. This can make solving trigonometric equations and verifying identities more manageable.
  • It simplifies expressions involving the sine of summed angles.
  • It is derived from the unit circle and geometric interpretations of sine and cosine.
Sine Function
The sine function helps us determine the vertical position of a point on the unit circle as it rotates around the circle. In trigonometry, the sine of an angle gives us the y-coordinate of the corresponding point on the unit circle. For instance, \( \sin \frac{\pi}{2} = 1 \), which tells us that the vertical position reaches its maximum.
  • It is periodic and oscillates between -1 and 1.
  • Sine is positive in the first and second quadrants of the coordinate system.
  • Each value repeats every \(2\pi\), which is the function's period.
The sine function is integral in wave motion, oscillations, and other applications, highlighting its importance in various scientific fields.
Cosine Function
The cosine function is another foundational trigonometric function. It corresponds to the horizontal position of a point on the unit circle. Thus, the cosine of an angle gives us the x-coordinate of the point. One of the useful characteristics of the cosine function is that \( \cos 0 = 1 \), showing where maximum horizontal displacement occurs.
  • The cosine function, like sine, is periodic with a period of \(2\pi\).
  • It varies from -1 to 1 as well, showing the same amplitude as the sine function.
  • Cosine is positive in the first and fourth quadrants, meaning the point lies on the right side of the unit circle.
The cosine function is used extensively in physics, engineering, and signal processing, mirroring the applications of the sine function. Its relationship with sine, particularly through identities like \(\sin(t + \frac{\pi}{2}) = \cos t\), shows the symmetry and interconnection between these two functions.

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Most popular questions from this chapter

Is cosine an even function, an odd function, or neither?

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