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Chapter 4: Problem 59

Find exact expressions for the indicated quantities. \(\cos (u-3 \pi)\)

Short Answer

Expert verified
\(\cos(u - 3\pi) = -\cos(u)\)

Step by step solution

01

Recall the cosine difference identity

Recall the cosine difference identity: \(\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)\) We will use this identity to find an expression for \(\cos(u - 3\pi)\).
02

Apply the identity to the given function

Let's apply the cosine difference identity to the given function: \(\cos(u - 3\pi) = \cos(u)\cos(3\pi) + \sin(u)\sin(3\pi)\)
03

Evaluate the trigonometric functions at the specific angles

Now we need to evaluate \(\cos(3\pi)\) and \(\sin(3\pi)\). Recall that \(3\pi = 2\pi + \pi\), and \(\cos(2\pi + \theta) = \cos(\theta)\) as well as \(\sin(2\pi + \theta) = \sin(\theta)\). Therefore, we get: \(\cos(3\pi) = \cos(\pi) = -1\) \(\sin(3\pi) = \sin(\pi) = 0\)
04

Substitute the evaluated functions into the expression

Substitute the values of \(\cos(3\pi)\) and \(\sin(3\pi)\) into the original expression: \(\cos(u - 3\pi) = \cos(u)(-1) + \sin(u)(0)\) Simplify the expression further: \(\cos(u - 3\pi) = -\cos(u) + 0 \) Finally, we have the exact expression for the given function: \(\boxed{\cos(u - 3\pi) = -\cos(u)}\)

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Key Concepts

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

Cosine Difference Identity
The cosine difference identity is one of the key identities in trigonometry. It relates the cosine of the difference of two angles to the cosines and sines of the individual angles. For two angles, \(A\) and \(B\), this identity is given by:
  • \(\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)\)
This formula helps simplify expressions involving cosines of angle differences. In our exercise, applying this identity to \(\cos(u - 3\pi)\) involves breaking it down into the cosine and sine components of \(u\) and \(3\pi\). This identity can be handy for solving problems in algebra and calculus where trigonometric expressions need to be simplified or evaluated.
Trigonometric Functions Evaluation
Evaluating trigonometric functions at specific angles is crucial to simplifying expressions using identities. Here, we needed to evaluate \(\cos(3\pi)\) and \(\sin(3\pi)\).
  • The angle \(3\pi\) is equivalent to \( \pi + 2\pi \), which brings us to an understanding of its trigonometric values.
  • We know \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\).
These evaluations are derived from the unit circle, which is a fundamental concept in trigonometry. For any \(\theta + 2\pi\), the cosine and sine values are the same as those for \(\theta\) due to the periodic nature of these functions.
Angle Transformation
When working with angles in trigonometry, understanding angle transformations is essential. Transforming angles typically involves comparing them to known values on the unit circle.
  • In the context of this exercise, transforming \(3\pi\) into \(\pi + 2\pi\) allows us to use the properties of periodicity in trigonometric functions.
  • The transformations leverage identities such as \(\cos(2\pi + \theta) = \cos(\theta)\) and \(\sin(2\pi + \theta) = \sin(\theta)\).
These concepts are vital for simplifying and solving trigonometric equations, especially when working with non-standard angles. This helps us find equivalent expressions that are easier to interpret or solve.

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Most popular questions from this chapter

The next two exercises emphasize that \(\cos ^{2} \theta\) does not ?qual \(\cos \left(\theta^{2}\right)\) For \(\theta=5\) radians, evaluate each of the following: (a) \(\cos ^{2} \theta\) (b) \(\cos \left(\theta^{2}\right)\)

Given that $$\cos 15^{\circ}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ Find exact expressions for the indicated quantities. [These values for \(\cos 15^{\circ}\) and \(\sin 22.5^{\circ}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sec 15^{\circ}\)

For Exercises \(29-34,\) assume the surface of the earth is a sphere with radius 3963 miles. The latitude of \(a\) point \(P\) on the earth's surface is the angle between the line from the center of the earth to \(P\) and the line from the center of the earth to the point on the equator closest to \(P\), as shown below for latitude \(40^{\circ} .\) Dallas has latitude \(32.8^{\circ}\) north. Find the radius of the circle formed by the points with the same latitude as Dallas.

Pretend that you are living in the time before calculators and computers existed, and that you have a table showing the cosines and sines of \(1^{\circ}, 2^{\circ}, 3^{\circ},\) and so on, up to the cosine and sine of \(45^{\circ}\). Explain how you would find the cosine and sine of \(71^{\circ}\), which are beyond the range of your table.

Suppose you need to find the height of a tall building. Standing 15 meters from the base of the building, you aim a laser pointer at the closest part of the top of the building. You measure that the laser pointer is \(7^{\circ}\) tilted from pointing straight up. The laser pointer is held 2 meters above the ground. How tall is the building?
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