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Chapter 4: Problem 65

Find exact expressions for the indicated quantities. \(\cos \left(\frac{\pi}{2}-u\right)\)

Short Answer

Expert verified
Using the co-function identity \(\cos\left(\frac{\pi}{2}-x\right) = \sin(x)\), we find that the exact expression for the given trigonometric function is \(\cos\left(\frac{\pi}{2}-u\right) = \sin(u)\).

Step by step solution

01

Identify the trigonometric function

The exercise asks us to find an exact expression for the trigonometric function \(\cos\left(\frac{\pi}{2}-u\right)\). Step 2: Apply the co-function identity
02

Apply the co-function identity

Using the co-function identity \(\cos\left(\frac{\pi}{2}-x\right) = \sin(x)\), we replace the angle \(\frac{\pi}{2}-u\) by \(x\): \[\cos\left(\frac{\pi}{2}-u\right) = \cos\left(x\right) \] Step 3: Find the expression for the trigonometric function
03

Find the expression for the trigonometric function

Now, using the identity \(\cos\left(\frac{\pi}{2}-x\right) = \sin(x)\), we get the exact expression for the trigonometric function as: \[\cos\left(\frac{\pi}{2}-u\right) = \sin(u) \] So, the exact expression for the given trigonometric function is \(\sin(u)\).

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Key Concepts

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

Co-function Identity
Co-function identities form a bridge between sine and cosine functions that can simplify trigonometric expressions. They are based on the relationships between complementary angles. A common co-function identity is \( \cos\left(\frac{\pi}{2} - x\right) = \sin(x) \). This means that the cosine of an angle is equal to the sine of its complementary angle.

These identities are useful in various mathematical contexts:
  • They simplify calculations by converting one function into another.
  • They're employed in solving trigonometric equations.
  • They aid in proving other mathematical theorems and identities.
This identity indicates that when you have \( \cos\left(\frac{\pi}{2} - u\right) \), it equals \( \sin(u) \). Understanding this equivalence is key to solving such problems efficiently.
Cosine Function
The cosine function measures the adjacent side over the hypotenuse in a right triangle. It's one of the primary trigonometric functions, essential for analyzing periodic phenomena like waves.

Here are some important aspects of the cosine function:
  • Periodicity: The cosine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) interval.
  • Symmetry: It's an even function, so \( \cos(-x) = \cos(x) \), reflecting symmetry over the y-axis.
  • Range and Domain: Its range is \([-1, 1]\), and the domain extends across all real numbers.
In the given problem, the focus is on using the co-function identity to transform a cosine expression into a sine function, taking advantage of these properties.
Sine Function
The sine function represents the opposite side over the hypotenuse in a right triangle. It's fundamentally important for understanding oscillatory behavior in physics and engineering.

Key features of the sine function include:
  • Periodicity: The sine function has a period of \( 2\pi \), so it repeats every \( 2\pi \) units.
  • Symmetry: It's an odd function, meaning \( \sin(-x) = -\sin(x) \), showing symmetry about the origin.
  • Range and Domain: The sine function also has a range of \([-1, 1]\) and is defined for all real numbers.
Incorporating the co-function identity, \( \cos\left(\frac{\pi}{2} - u\right) = \sin(u) \), demonstrates how the sine function can emerge directly from a cosine expression, showcasing the interconnected nature of these trigonometric functions.

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Most popular questions from this chapter

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\cos u\)

Show that $$\cos (\pi-\theta)=-\cos \theta$$ for every angle \(\theta\).

Find the smallest number \(x\) such that $$ \cos \left(e^{x}+1\right)=0. $$

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\cos \frac{25 \pi}{12}\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\sin v\)
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