Problem 114 - Precalculus : Building Concepts and Connections

Question

Suppose \(t\) is in \(\left(0, \frac{\pi}{2}\right) .\) Express \(\sin \left(t+\frac{\pi}{2}\right)\) in terms of sin \(t .\) (Hint: It is helpful to sketch a figure.)

Step-by-Step Solution

Answer

The expression \(\sin\left(t+\frac{\pi}{2}\right)\) is equivalent to \(\cos(t)\), or alternatively, \(\sin\left(\frac{\pi}{2} - t\right)\).

Step 1: Apply the angle addition identity for sine

The angle addition identity for sine is \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\). Plugging in our specific values, we get \(\sin\left(t+\frac{\pi}{2}\right) = \sin(t)\cos\left(\frac{\pi}{2}\right) + \cos(t)\sin\left(\frac{\pi}{2}\right)\).

Step 2: Use the fact that sine and cosine of \(\frac{\pi}{2}\) are known values

\(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\). Substituting these values into our expression from Step 1, we get \(\sin\left(t+\frac{\pi}{2}\right) = \sin(t)*0 + \cos(t)*1 = \cos(t)\).

Step 3: Remembering the relationship between sine and cosine

For any real number \(t\), there is the identity \(\sin\left(\frac{\pi}{2} - t\right) = \cos(t)\). Thus we can write \(\sin\left(t+\frac{\pi}{2}\right) = \cos(t) = \sin\left(\frac{\pi}{2} - t\right)\).

Key Concepts

Angle Addition Identity

The angle addition identity is foundational in trigonometry. It allows us to find the sine, cosine, or tangent of the sum of two angles using simpler existing values. Specifically, for sine, the identity is given by:

• \[\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\]

This formula is particularly useful when we need to determine the sine of an angle expressed as a sum of two angles, where only their individual sine and cosine values are known. In the case of the exercise, this identity helps us find \(\sin(t + \frac{\pi}{2})\), simplifying it to terms we better understand.

To practice, it's helpful to draw out these angles on the unit circle. Visualizing can aid in understanding how the components of sine and cosine interact when angles are added.

Sine Function

The sine function is one of the primary functions in trigonometry. It relates a point on a unit circle to its y-coordinate, representing the ratio of the length of the opposite side of an angle in a right triangle to its hypotenuse. The function is defined as:

• \(\sin(\theta)\), where \(\theta\) is the angle.

The sine function has a range of [-1, 1], and is periodic with a period of \(2\pi\).

In the context of the problem, understanding the sine function helps as we transform \(\sin(t + \frac{\pi}{2})\) into terms of \(\cos(t)\), highlighting the relationship between sine and cosine. This transformation underscores the complementary nature of these trigonometric functions.

Cosine Function

The cosine function complements the sine function by calculating the x-coordinate of a point on the unit circle. In trigonometric terms, it represents the ratio of the length of the adjacent side of an angle in a right triangle to the hypotenuse. The function is described as:

• \(\cos(\theta)\), where \(\theta\) is the angle.

The range of cosine is also [-1, 1] and it mirrors the sine function in periodicity with a period of \(2\pi\).

In the provided exercise, when we find that \(\sin(t + \frac{\pi}{2}) = \cos(t)\), it’s a direct application of how angles shifted by \(\frac{\pi}{2}\) translate sine functions into cosine functions. The identity presented in the solution reflects this shift and equates the two functions when rotated by this critical angle.

\(\pi/2\) Angle

The angle \(\frac{\pi}{2}\) radians, equivalent to 90 degrees, holds significant importance in trigonometry. It’s a critical angle where key trigonometric values are defined:

• \(\sin\left(\frac{\pi}{2}\right) = 1\)
• \(\cos\left(\frac{\pi}{2}\right) = 0\)

These values are indispensable when applying angle addition identities like in our exercise.

Understanding these values simplifies solving trigonometric expressions like \(\sin(t + \frac{\pi}{2})\), where substituting \(\cos\left(\frac{\pi}{2}\right)\) and \(\sin\left(\frac{\pi}{2}\right)\) streamlines calculations. Knowing these key angles and how they interact with sine and cosine functions provides a deeper grasp of the symmetry and periodic nature inherent in trigonometric concepts.