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Chapter 6: Problem 29

Use the given information to determine the values of the remaining five trigonometric functions. (The angles are assumed to be acute angles. ) $$\cos B=4 / 7$$

Short Answer

Expert verified
\( \sin B = \frac{\sqrt{33}}{7}, \tan B = \frac{\sqrt{33}}{4}, \csc B = \frac{7\sqrt{33}}{33}, \sec B = \frac{7}{4}, \cot B = \frac{4\sqrt{33}}{33} \).

Step by step solution

01

Understand Given Information

We are given that \( \cos B = \frac{4}{7} \). Since \( B \) is an acute angle, \( \cos B \) gives us the ratio of the adjacent side to the hypotenuse in a right triangle.
02

Calculate \( \sin B \)

Using the Pythagorean identity: \( \sin^2 B + \cos^2 B = 1 \). Plug in \( \cos B = \frac{4}{7} \):\[\sin^2 B + \left(\frac{4}{7}\right)^2 = 1\]\[\sin^2 B + \frac{16}{49} = 1\]\[\sin^2 B = \frac{33}{49}\]\[\sin B = \sqrt{\frac{33}{49}} = \frac{\sqrt{33}}{7}\]
03

Calculate \( \tan B \)

The tangent of angle \( B \) is the ratio of \( \sin B \) to \( \cos B \):\[\tan B = \frac{\sin B}{\cos B} = \frac{\frac{\sqrt{33}}{7}}{\frac{4}{7}} = \frac{\sqrt{33}}{4}\]
04

Calculate \( \csc B \)

The cosecant is the reciprocal of the sine:\[\csc B = \frac{1}{\sin B} = \frac{7}{\sqrt{33}}\]Rationalize the denominator:\[\csc B = \frac{7\sqrt{33}}{33}\]
05

Calculate \( \sec B \)

The secant is the reciprocal of the cosine:\[\sec B = \frac{1}{\cos B} = \frac{7}{4}\]
06

Calculate \( \cot B \)

The cotangent is the reciprocal of the tangent:\[\cot B = \frac{1}{\tan B} = \frac{4}{\sqrt{33}}\]Rationalize the denominator:\[\cot B = \frac{4\sqrt{33}}{33}\]

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Key Concepts

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

Pythagorean Identity
The Pythagorean identity is a fundamental relationship in trigonometry. It states that in a right triangle, the square of the sine of an angle plus the square of the cosine of the angle equals one: \[\sin^2 \theta + \cos^2 \theta = 1\]This identity derives from the Pythagorean theorem, which relates the three sides of a right triangle. If you know the cosine or sine of an angle, you can always find the other using this identity. For example, if you know \(\cos B = \frac{4}{7}\), substitute it into the identity:
  • \(\sin^2 B + \left(\frac{4}{7}\right)^2 = 1\)
  • Solve for \(\sin^2 B\)
  • Get \(\sin B = \frac{\sqrt{33}}{7}\)
Learning this identity helps simplify complex trigonometric problems and is essential for further studies in mathematics.
Reciprocal Identities
Reciprocal identities are used in trigonometry to find the reciprocals of the fundamental trigonometric functions: sine, cosine, and tangent. These identities are particularly useful when calculating lesser-known trigonometric functions like cosecant, secant, and cotangent. They are defined as:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
For instance, when given \(\cos B = \frac{4}{7}\), you can find \(\sec B\) using the reciprocal identity: \[\sec B = \frac{1}{\cos B} = \frac{7}{4}\]Using these relationships, you can express all trigonometric functions in terms of sine, cosine, and tangent.
Right Triangle
A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This unique characteristic forms the basis of many trigonometric principles and calculations.
  • The side opposite the right angle is the hypotenuse, the longest side.
  • The other two sides are called the adjacent and opposite sides, relative to the angle of interest.
  • Understanding these sides helps determine trigonometric functions.
In the given problem, knowing \(\cos B = \frac{4}{7}\) implies that for angle \(B\), the adjacent side in a right triangle is 4 units long, and the hypotenuse is 7 units long. This understanding is crucial when using trigonometric ratios to find missing information about the triangle.
Trigonometric Ratios
Trigonometric ratios relate the angles of a triangle to the lengths of its sides. These ratios are essential for solving problems involving right triangles, as well as in various applications such as physics and engineering.
  • \(\sin \theta\) is the ratio of the opposite side to the hypotenuse.
  • \(\cos \theta\) is the ratio of the adjacent side to the hypotenuse.
  • \(\tan \theta\) is the ratio of the opposite side to the adjacent side.
Given \(\cos B = \frac{4}{7}\), we already know the adjacent side and hypotenuse. By using the Pythagorean identity, you can find \(\sin B = \frac{\sqrt{33}}{7}\). Then, the tangent \(\tan B\) is calculated as:\[\tan B = \frac{\sin B}{\cos B} = \frac{\frac{\sqrt{33}}{7}}{\frac{4}{7}} = \frac{\sqrt{33}}{4}\]Trigonometric ratios enable us to connect angles with side lengths, providing a powerful tool for solving geometric problems.

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Most popular questions from this chapter

Four functions \(S, C, T,\) and \(D\) are defined as follows: \(\left.\begin{array}{l}S(\theta)=\sin \theta \\ C(\theta)=\cos \theta \\\ T(\theta)=\tan \theta \\ D(\theta)=2 \theta\end{array}\right\\} \quad 0^{\circ}<\theta<90^{\circ}\) In each case, use the values to decide if the statement is true or false. A calculator is not required. $$S\left(45^{\circ}\right)-C\left(45^{\circ}\right)=0$$

This exercise is adapted from a problem that appears in the classic text \(A\) Treatise on Plane and Advanced Trigonometry, 7th ed., by E. W. Hobson (New York: Dover Publications, 1928 ). (The first edition of the book was Oublished by Cambridge University Press in \(1891 .)\) Given: \(A, B,\) and \(C\) are acute angles such that \(\cos A=\tan B \quad \cos B=\tan C \quad \cos C=\tan A\) Prove: \(\sin A=\sin B=\sin C=2 \sin 18^{\circ}\) Follow steps (a) through (e) to obtain this result. (a) In each of the three given equations, use the identity \(\tan \theta=(\sin \theta) /(\cos \theta)\) so that the equations contain only sines and cosines. (b) In each of the three equations obtained in part (a), square both sides. Then use the identity \(\sin ^{2} \theta=1-\cos ^{2} \theta\) so that each equation contains only the cosine function. (c) For ease in writing, replace \(\cos ^{2} A, \cos ^{2} B,\) and \(\cos ^{2} C\) by \(a, b,\) and \(c,\) respectively. Now you have a system of three equations in the three unknowns \(a, b,\) and \(c .\) Solve for \(a, b,\) and \(c\) (d) Using the results in part (c), show that $$\sin A=\sin B=\sin C=\sqrt{\frac{3-\sqrt{5}}{2}}$$ (e) From Exercise \(54(\mathrm{f})\) we know that \(\sin 18^{\circ}=\) \((\sqrt{5}-1) / 4 .\) Show that the expression obtained in part (d) is equal to twice this expression for \(\sin 18^{\circ}\) This completes the proof. (Use the fact that two nonnegative quantities are equal if and only if their squares are equal.)

Use a calculator to evaluate \(\sec \theta, \csc \theta,\) and cot \(\theta\) for the given value of \(\theta .\) Round the answers to two decimal places. $$-9$$

Let $$\begin{aligned}S(\theta) &=\sin \theta \\\C(\theta) &=\cos \theta \\\L(x) &=\ln x \end{aligned} \quad 0^{\circ} \leq \theta \leq 360^{\circ}$$ What is the domain of the function \((L \circ S)(\theta) ?\)

Use the given information to express the remaining five trigonometric values of the angle \(\theta\) in terms of \(x .\) (Rationalize any denominators containing radicals.) \(\sin \theta=x / 2\) (a) if \(0^{\circ}<\theta<90^{\circ}\) (b) if \(180^{\circ}<\theta<270^{\circ}\)
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