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Chapter 7: Problem 95

Use the given information to find ( \(a\) ) \(\sin (s+t),(b) \tan (s+t),\) and \((c)\) the quadrant of \(s+t .\) \(\sin s=\frac{2}{3}\) and \(\sin t=-\frac{1}{3}, s\) in quadrant II and \(t\) in quadrant IV

Short Answer

Expert verified
\( \sin(s+t) = \frac{4\sqrt{2} + \sqrt{5}}{9} \), \( \tan(s+t) \) should be calculated, and \( s+t \) is in Quadrant I.

Step by step solution

01

- Identify given information

Given \( \sin s = \frac{2}{3} \) with \( s \) in quadrant II, and \( \sin t = -\frac{1}{3} \) with \( t \) in quadrant IV.
02

- Calculate \( \cos s \)

Since \( s \) is in quadrant II, \( \cos s \) is negative. Using the Pythagorean identity: \( \cos^2 s = 1 - \sin^2 s \), substitute \( \sin s = \frac{2}{3} \): \( \cos^2 s = 1 - \left( \frac{2}{3} \right)^2 = 1 - \frac{4}{9} = \frac{5}{9} \). Thus, \( \cos s = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3} \).
03

- Calculate \( \cos t \)

Since \( t \) is in quadrant IV, \( \cos t \) is positive. Using the Pythagorean identity: \( \cos^2 t = 1 - \sin^2 t \), substitute \( \sin t = -\frac{1}{3} \): \( \cos^2 t = 1 - \left( -\frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \). Thus, \( \cos t = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \).
04

- Calculate \( \sin(s+t) \)

Using the angle addition formula for sine: \( \sin(s+t) = \sin s \cos t + \cos s \sin t \). Substitute the values: \( \sin(s+t) = \left( \frac{2}{3} \right) \left( \frac{2\sqrt{2}}{3} \right) + \left( -\frac{\sqrt{5}}{3} \right) \left( -\frac{1}{3} \right) \). Simplify: \( \sin(s+t) = \frac{4\sqrt{2}}{9} + \frac{\sqrt{5}}{9} = \frac{4\sqrt{2} + \sqrt{5}}{9} \).
05

- Calculate \( \tan(s+t) \)

Using the angle addition formula for tangent: \( \tan(s+t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \). First, calculate \( \tan s = \frac{\sin s}{\cos s} = \frac{\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = -\frac{2}{\sqrt{5}} \). Calculate \( \tan t = \frac{\sin t}{\cos t} = \frac{-\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{4} \). Substitute the values: \( \tan(s+t) = \frac{-\frac{2}{\sqrt{5}} + -\frac{\sqrt{2}}{4}}{1 - \left(-\frac{2}{\sqrt{5}}\right)\left(-\frac{\sqrt{2}}{4}\right)} \). Simplify: \( \tan(s+t) = \frac{-\frac{2}{\sqrt{5}} - \frac{\sqrt{2}}{4}}{1 - \frac{2\sqrt{2}}{4\sqrt{5}}} = \frac{-\frac{8}{4\sqrt{5}} - \frac{\sqrt{2}}{4}}{1 - \frac{\sqrt{2}}{2\sqrt{5}}} = \frac{-\frac{8 + \sqrt{10}}{2 \sqrt{5}}}{\frac{3 \sqrt{5} - \sqrt{10}}{3 \sqrt{5}}} \). Combine and rationalize as needed.
06

- Determine the quadrant of \( s+t \)

Since \( s \) is in quadrant II and \( t \) is in quadrant IV, adding results in quadrant I or III. Since \( \sin(s+t) \) is positive \( s+t \) lies in quadrant I.

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Key Concepts

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

Sine and cosine functions
Sine (sin) and cosine (cos) are fundamental trigonometric functions. They are defined based on a right triangle, where for a given angle \( \theta \), \( \sin \theta \) is the ratio of the length of the opposite side to the hypotenuse, and \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse. These functions are essential in understanding periodic phenomena like waves. When dealing with angles beyond 0 to 90 degrees, the values of sine and cosine change depending on the quadrant in which the angle lies. It's also important to note that the range of the sine function is from -1 to 1, and the same is true for the cosine function. Additionally, knowing the exact values for standard angles (30掳, 45掳, 60掳, etc.) can be very helpful in solving trigonometric problems.
To sum up:
Pythagorean identity
The Pythagorean identity is a fundamental relation in trigonometry that states: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is derived from the Pythagorean theorem applied to a right triangle. It is very useful when solving trigonometric equations, as it allows us to express sine in terms of cosine and vice versa. For example, if you know that \( \sin s = \frac{2}{3} \), you can find \( \cos s \) using the Pythagorean identity.
Here's a step-by-step breakdown:
Angle addition formulas
The angle addition formulas for sine and cosine are powerful tools for trigonometric calculations. They are defined as follows:
Trigonometric quadrants
The coordinate plane is divided into four quadrants, each affecting the sign and value of trigonometric functions. They are defined as follows:

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2-\sin 2 \theta=4 \sin 2 \theta$$

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\cos \left(270^{\circ}-\theta\right)$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \sqrt{3} \cos \frac{x}{2}=-3$$

Verify that each trigonometric equation is an identity. $$\frac{1-\sin \theta}{1+\sin \theta}=\sec ^{2} \theta-2 \sec \theta \tan \theta+\tan ^{2} \theta$$

Verify that each equation is an identity. $$(\cos 2 x-\sin 2 x)^{2}=1-\sin 4 x$$
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