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Chapter 5: Problem 48

Complete the identity. $$\cos \left(90^{\circ}-\theta\right)=\square$$

Short Answer

Expert verified
The completed identity is \( \cos(90^{\circ}-\theta)=\sin \theta \)

Step by step solution

01

Identify the pattern

This formula appears like a part of the co-function identities which states that \( \cos(90^{\circ} - \theta) = \sin \theta \). This identity results from the fact that cosine and sine are co-functions.
02

Apply the co-function identity

Applying the co-function identities, we can ascertain that \( \cos(90^{\circ}-\theta) \) equals \( \sin \theta \), thus completing the identity.

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Key Concepts

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

Co-Function Identities
When it comes to trigonometry, co-function identities are incredibly useful. They show a unique relationship between the sine and cosine functions. These equations reveal how the values of sine and cosine can swap their roles depending on the angles used.

One of the key co-function identities is:
  • \( \,\cos(90^{\circ} - \theta) = \sin \theta \,\)
  • \( \,\sin(90^{\circ} - \theta) = \cos \theta \,\)
In these identities, if the function is cosine, you subtract the angle from 90 degrees to change it to sine. And vice versa. It's all about understanding how these two functions relate to each other. Once you know this, many trigonometric problems become a lot simpler to solve.
Cosine
Cosine is a fundamental trigonometric function and abbreviated as \( \cos \). It is defined for a right-angled triangle as the ratio of the length of the adjacent side to the length of the hypotenuse.

In mathematical terms, for a given angle \( \theta \):\[\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}.\]Cosine has properties that relate it to the unit circle, where the angle \( \theta \) is measured from the positive x-axis.
  • The range of the cosine function is between -1 and 1.
  • Cosine has a period of \(360^{\circ} \) or \( 2\pi \) in radians.
  • \(\cos(90^{\circ}) = 0\) , meaning when \( \theta = 90^{\circ} \), cosine becomes zero.
These characteristics help in solving various trigonometric equations and identities.
Sine
The sine function, denoted as \( \sin \), is one of the most essential trigonometric functions. For a right-angled triangle, sine is the ratio of the opposite side's length to the hypotenuse's length.

This can be expressed as: \[\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}.\]Like cosine, sine is also connected to the unit circle. It represents the y-coordinate of a point on the unit circle.
  • The sine function ranges between -1 and 1 for all angles.
  • It's periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.
  • \(\sin(90^{\circ}) = 1\), the highest point on the unit circle.
Understanding the sine function is crucial for mastering trigonometric problems.
Angle Transformations
Angle transformations involve changing or transforming one angle into another, often using special formulas in trigonometry. They are crucial for solving trigonometric identities and equations.

Basic angle transformations can include converting from degrees to radians or vice versa, and they help us analyze angles in different units:
  • To convert degrees to radians, use the formula: \( \,\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\)
  • To convert radians to degrees, the formula is: \( \,\text{Degrees} = \frac{180}{\pi} \times \text{Radians}\)
In the context of co-function identities, angle transformations help us understand how an expression like \( \,\cos(90^{\circ} - \theta)\) can be evaluated by transforming it to \( \,\sin \theta\). This comes from the idea that co-functions are complementary angles that add up to 90 degrees.

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Most popular questions from this chapter

True or False Determine whether the statement is true or false. Justify your answer. Simple harmonic motion does not involve a damping factor.

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=5 \pi / 2$$

Determine whether the statement is true or false. Justify your answer. The graph of the function given by \(g(x)=\sin (x+2 \pi)\) translates the graph of \(f(x)=\sin x\) one period to the right.

The table shows the average daily high temperatures (in degrees Fahrenheit) for Quillayute, Washington, \(Q\) and Chicago, Illinois, \(C\) for month \(t,\) with \(t=1\) corresponding to January. $$\begin{array}{c|c|c} \text { Month, } & \text { Quillayute, } & \text { Chicago, } \\ t & Q & C \\ \hline 1 & 47.1 & 31.0 \\ 2 & 49.1 & 35.3 \\ 3 & 51.4 & 46.6 \\ 4 & 54.8 & 59.0 \\ 5 & 59.5 & 70.0 \\ 6 & 63.1 & 79.7 \\ 7 & 67.4 & 84.1 \\ 8 & 68.6 & 81.9 \\ 9 & 66.2 & 74.8 \\ 10 & 58.2 & 62.3 \\ 11 & 50.3 & 48.2 \\ 12 & 46.0 & 34.8 \end{array}$$ (a) \(\mathrm{A}\) model for the temperature in Quillayute is given by $$Q(t)=57.5+10.6 \sin (0.566 x-2.568)$$ Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data and the model for the temperatures in Quillayute in the same viewing window. How well does the model fit the data? (c) Use the graphing utility to graph the data and the model for the temperatures in Chicago in the same viewing window. How well does the model fit the data? (d) Use the models to estimate the average daily high temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.

Your classmate uses a calculator to evaluate \(\tan (\pi / 2)\) and gets a result of 0.0274224385 Describe the error.
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