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Chapter 6: Problem 48

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin \left(\frac{\pi}{2}+t\right)=-\cos t$$

Short Answer

Expert verified
Answer: The given equation is definitely not an identity.

Step by step solution

01

Graph the left side of the equation

First, we need to graph the function $$y = \sin\left(\frac{\pi}{2} + t\right)$$. This graph will show us how the sine function behaves when shifted to the left by $$\frac{\pi}{2}$$ on the horizontal axis.
02

Graph the right side of the equation

Next, we need to graph the function $$y = -\cos(t)$$. This graph will show us the behavior of the cosine function when multiplied by -1, which results in the reflection of the graph about the x-axis.
03

Compare the two graphs

Now we need to compare the two graphs we produced in Steps 1 and 2. If they are identical, it is possible that the equation is an identity.
04

Analyze their properties

To further determine if the equation is an identity, we can recall the properties of trigonometric functions. The equation $$\sin(\frac{\pi}{2} + t) = -\cos(t)$$ can be rewritten using the cofunction identity $$\sin\left(\frac{\pi}{2} + t\right) = \cos\left(\frac{\pi}{2} - t\right)$$. Now, we can subtract t from both sides of the equation and see if the result simplifies to the original equation: $$\cos\left(\frac{\pi}{2} - t\right) + t = \cos\left(\frac{\pi}{2} - t\right) - t$$ Since $$\cos\left(\frac{\pi}{2} - t\right)$$ is a function of t and cannot be added to a scalar, the equation cannot be simplified to the original equation.
05

Conclusion

After comparing the graphs of the left side and the right side of the equation and analyzing their properties, we can conclude that the given equation is definitely not an identity because the graphs of the two sides are not identical and their properties do not lead to a valid simplification.

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Key Concepts

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

Graphing Trigonometric Functions
Graphing trigonometric functions like sine (\(y = \sin(t)\)) and cosine (\(y = \cos(t)\)) is essential to understanding their behavior. The graphs allow us to see the repeating patterns or the "waves" these functions create. Both of these functions have a period of \(2\pi\), meaning they repeat every \(2\pi\) units along the x-axis.
To graph a trigonometric function effectively:
  • Identify the function type (sine or cosine).
  • Determine any transformations such as shifts or reflections.
  • Calculate key points such as peaks, troughs, and zero crossings.
  • Sketch the wave ensuring the correct period and amplitude are visible.
Observing these graphs visually makes it easier to compare different trigonometric expressions to determine if they may be identities.
Sine and Cosine Function Transformation
Transformations of the sine and cosine functions change their graphical representation, making it easier to match different trigonometric expressions visually. Transformations include shifts, reflections, and stretches:
  • Shifts: Moving the graph horizontally or vertically, for instance, \(y = \sin(t + \pi/2)\) shifts the sine graph to the left by \( \pi/2 \) units.
  • Reflections: Multiplying by -1 reflects the graph across an axis, as in \(y = -\cos(t)\), reflecting the cosine graph across the x-axis.
  • Stretches: Changing the amplitude or period, like \(y = 2\sin(t)\) makes the sine wave twice as tall.
Understanding these transformations is vital for recognizing equivalent trigonometric equations and aids in determining if they can be identities. By applying transformations, one can superimpose graphs for comparison, checking for identity.
Cofunction Identities
Cofunction identities relate trigonometric functions to each other through complementary angles. They are a critical part of simplifying and understanding trigonometric expressions. One common cofunction identity is:\[\sin\left(\frac{\pi}{2} + t\right) = \cos(t)\]This identity shows how a sine function can transform into a cosine function under specific angle adjustments.
Using cofunction identities:
  • Allows simplification of trigonometric expressions.
  • Facilitates the comparison of different trigonometric equations.
  • Enhances problem-solving by revealing relationships between functions.
In our analysis task, the role of cofunction identities was to attempt to equate both sides of the equation given in the exercise to prove whether it could actually be an identity. Understanding these identities helps clarify when it is plausible or implausible for one expression to equal another through graphing and algebraic manipulation.

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Most popular questions from this chapter

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$f(t)=-2 \sin (3 t-\pi)$$

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 1 minute, 2 rpm

A pendulum swings uniformly back and forth, taking 2 seconds to move from the position directly above point \(A\) to the position directly above point \(B\). (Check your book to see image) The distance from \(A\) to \(B\) is 20 centimeters. Let \(d(t)\) be the horizontal distance from the pendulum to the (dashed) center line at time \(t\) seconds (with distances to the right of the line measured by positive numbers and distances to the left by negative ones). Assume that the pendulum is on the center line at time \(t=0\) and moving to the right. Assume that the motion of the pendulum is simple harmonic motion. Find the rule of the function \(d(t)\)

On the basis of the results of Exercises \(37-42,\) under what conditions on the constants \(a, k, h, d, r, s\) does it appear that the graph of $$f(t)=a \sin (k t+h)+d \cos (r t+s)$$ coincides with the graph of the function $$g(t)=A \sin (b t+c) ?$$

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$h(t)=\frac{1}{2} \cos \left(\frac{\pi}{2} t-\frac{\pi}{8}\right)+1$$
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