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Chapter 5: Problem 38

Find the exact values of \(\sin 2 u, \cos 2 u,\) and tan \(2 u\) using the double- angle formulas. $$\cos u=-\frac{4}{5}, \quad \frac{\pi}{2}

Short Answer

Expert verified
\(\sin 2u = -24/25\), \(\cos 2u = 7/25\), and \(\tan 2u = -24/7\)

Step by step solution

01

Determine the Sign of Sin u

Given that \(\cos u = -4/5\) and \(\pi/2 < u < \pi\), u lies in the second quadrant. In the second quadrant sine is positive. So \(\sin u\) is positive.
02

Find Sin u

Use the Pythagorean identity \(\sin^2 u = 1 - \cos^2 u\). Substituting \(\cos u = -4/5\) into the formula we get \(\sin u = \sqrt{1 - (-4/5)^2} = \sqrt{1 - 16/25} = \sqrt{9/25} = 3/5\). Hence, \(\sin u = 3/5\). Note that we only consider the positive root as we have determined \(\sin u\) to be positive.
03

Find Sin 2u

Using the double-angle formula \(\sin 2u = 2 \sin u \cos u\). Substituting \(\sin u = 3/5\) and \(\cos u = -4/5\) gives \(\sin 2u = 2 * 3/5 * -4/5 = -24/25\). Hence, \(\sin 2u = -24/25\).
04

Find Cos 2u

Using the double-angle formula \(\cos 2u = \cos^2 u - \sin^2 u\). Substituting \(\sin u = 3/5\) and \(\cos u = -4/5\) gives \(\cos 2u = (-4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25\). Hence, \(\cos 2u = 7/25\).
05

Find Tan 2u

Use the relationship \(\tan 2u = \sin 2u/ \cos 2u\). Substituting \(\sin 2u = -24/25\) and \(\cos 2u = 7/25\) gives \(\tan 2u = -24/25 / 7/25 = -24/7\). Hence, \(\tan 2u = -24/7\).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are fundamental equations involving trigonometric functions like sine, cosine, and tangent. These identities help relate the angles and sides of right triangles.
Among the most useful are the double-angle formulas, which express trigonometric functions of double angles.
  • The \(\sin 2u = 2 \sin u \cos u\) formula allows us to compute the sine of a double angle using the sine and cosine of the original angle, \(u\).
  • Likewise, the \(\cos 2u = \cos^2 u - \sin^2 u\) formula helps us calculate the cosine of a double angle.
  • For the tangent of a double angle, the formula is \(\tan 2u = \frac{\sin 2u}{\cos 2u}\).
These formulas simplify the process of finding trigonometric values for doubled angles, essential in solving various trigonometric problems.
Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. This circle is crucial in trigonometry as it provides a geometric representation of trigonometric functions.
In the unit circle:
  • The angle \(u\) is measured from the positive x-axis, counterclockwise being positive, and clockwise being negative.
  • The coordinates \((x, y)\) on the circle are \((\cos u, \sin u)\).
Because of the unit circle, every trigonometric problem involving angles can also be visualized geometrically.
It helps in understanding the relationship between the angle's size and its sine and cosine values.
Pythagorean Identity
The Pythagorean identity is one of the most important trigonometric identities. It relates the square of the sine and cosine of a single angle. The formula is given by:
  • \(\sin^2 u + \cos^2 u = 1\)
This identity comes from the unit circle. The radius of the unit circle is \(1\), and any point \((\cos u, \sin u)\) on the circle satisfies the equation of a circle of radius 1.
Use this identity to find one trigonometric function if you know the other, such as finding \(\sin u\) when \(\cos u\) is given.
It is also vital for simplifying expressions and equations in trigonometry.
Second Quadrant Angles
Angles in the second quadrant range between \(\frac{\pi}{2}\) and \(\pi\). Understanding the sign of trigonometric functions in this quadrant is essential.
  • In the second quadrant, \(\sin u\) is positive because sine increases from 0 to 1.
  • \(\cos u\) is negative because cosine decreases from 1 to -1 as angles move from the first to the second quadrant.
By understanding the quadrant, you can determine the signs of sine, cosine, and tangent.
For instance, knowing that \(u\) is in the second quadrant tells us immediately \(\sin u\) is positive, and \(\cos u\) is negative. This information is crucial when using identities like the double-angle formulas, because the sign affects the resulting values.

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Most popular questions from this chapter

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