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Chapter 8: Problem 3

If \(\tan (\mathrm{A}+\mathrm{B})=\sqrt{3}\) and \(\tan (\mathrm{A}-\mathrm{B})=\frac{1}{\sqrt{3}} ; 0^{\circ}<\mathrm{A}+\mathrm{B} \leq 90^{\circ} ; \mathrm{A}>\mathrm{B}\), find \(\mathrm{A}\) and \(\mathrm{B}\).

Short Answer

Expert verified
\(A = 60^\circ\) and \(B = 30^\circ\).

Step by step solution

01

Apply Tangent Addition Formula

Use the tangent addition formula: \(\tan(\phi + \theta) = \frac{\tan(\phi) + \tan(\theta)}{1 - \tan(\phi)\tan(\theta)}\). Let \(\phi = A\) and \(\theta = B\), and for the first condition \(\tan(A+B) = \sqrt{3}\), set up the equation using the formula.
02

Apply Tangent Subtraction Formula

Use the tangent subtraction formula: \(\tan(\phi - \theta) = \frac{\tan(\phi) - \tan(\theta)}{1 + \tan(\phi)\tan(\theta)}\). Again, let \(\phi = A\) and \(\theta = B\), and for the second condition \(\tan(A-B) = \frac{1}{\sqrt{3}}\), set up this equation using the formula.
03

Solve the System of Equations

Solve the two equations obtained from Step 1 and Step 2 simultaneously to find the values of \(\tan(A)\) and \(\tan(B)\). This involves algebraic manipulation to isolate \(\tan(A)\) and \(\tan(B)\).
04

Determine Angles A and B

Using the values of \(\tan(A)\) and \(\tan(B)\), find the angles \(A\) and \(B\) by taking the arctangent of each. This must be done within the given range \(0^\circB\).
05

Verify Your Solution

It's important to plug your values of \(A\) and \(B\) back into the original equations to make sure they satisfy both \(\tan(A+B)=\sqrt{3}\) and \(\tan(A-B)=\frac{1}{\sqrt{3}}\).

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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 equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equation are defined. These identities are crucial for simplifying expressions, solving trigonometric equations, and transforming one trigonometric form into another.

One of the most frequently used set of identities are the tangent identities, which include the tangent addition and subtraction formulas. For example, the formula \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\) is used to find the tangent of the sum of two angles. Understanding these identities allows you to solve complex trigonometric problems and find angles or lengths in geometry. They are foundational tools that every student of trigonometry should master.
System of Equations
A system of equations is a collection of two or more equations with the same set of unknowns. In the realm of trigonometry, solving a system of equations often requires finding the values of trigonometric functions that satisfy all equations in the system simultaneously.

When working with trigonometric systems, you can expect to deal with nonlinear equations, as shown in the tangent addition and subtraction formulas. To solve these, you can utilize substitution or elimination methods, and sometimes it's necessary to apply trigonometric identities to simplify the equations first. Understanding how to manipulate and solve a system of equations is essential for finding precise solutions to trigonometric problems, such as calculating angles or side lengths in triangles.
Angle Measurement
Angle measurement is fundamental in trigonometry and geometry. Angles are typically measured in degrees or radians. One complete revolution, which is a full circle, is equal to \(360^\circ\) in the degree system, and \(2\pi\) radians in the radian system.

When solving trigonometric problems, accurate angle measurement is crucial. It defines the specific value of trigonometric functions that are periodic in nature. A deep understanding of how angles are measured helps students navigate through problems that require converting between different measurement units or determining the angles in various geometric shapes. The angle measurement system also ties into understanding inverse trigonometric functions, as the outcomes are reflected in the unit of angles.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverses of the regular trigonometric functions. They allow us to find the angle that corresponds to a given trigonometric ratio. For instance, the inverse of the tangent function is the arctangent, denoted by \(\arctan\) or \(\tan^{-1}\).

These functions are crucial when we need to determine the measure of an angle from its sine, cosine, or tangent value. When using inverse trigonometric functions, especially in the context of solving equations, it's important to consider the range over which these functions are defined to ensure that the angle we find is within the correct interval, as there are infinitely many angles that can have the same trigonometric value due to the periodic nature of these functions.

For example, when calculating the angles \(A\) and \(B\) given their tangent values, we use the arctangent function within the specified range to ensure a unique solution.

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Most popular questions from this chapter

Evaluate the following: (i) \(\sin 60^{\circ} \cos 30^{\circ}+\sin 30^{\circ} \cos 60^{\circ}\) (ii) \(2 \tan ^{2} 45^{\circ}+\cos ^{2} 30^{\circ}-\sin ^{2} 60^{\circ}\) (iii) \(\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}\) (iv) \(\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}\) (v) \(\frac{5 \cos ^{2} 60^{\circ}+4 \sec ^{2} 30^{\circ}-\tan ^{2} 45^{\circ}}{\sin ^{2} 30^{\circ}+\cos ^{2} 30^{\circ}}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) \((\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta}\) (ii) \(\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A\) (iii) \(\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta\) [Hint : Write the expression in terms of \(\sin \theta\) and \(\cos \theta]\) (iv) \(\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A} \quad[\) Hint \(:\) Simplify LHS and RHS separately] (v) \(\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\operatorname{cosec} A+\cot A\), using the identity \(\operatorname{cosec}^{2} A=1+\cot ^{2} A\). (vi) \(\sqrt{\frac{1+\sin \mathrm{A}}{1-\sin \mathrm{A}}}=\sec \mathrm{A}+\tan \mathrm{A}\) (vii) \(\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta\) (viii) \((\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A\) (ix) \((\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}\) [Hint : Simplify LHS and RHS separately] (x) \(\left(\frac{1+\tan ^{2} \mathrm{~A}}{1+\cot ^{2} \mathrm{~A}}\right)=\left(\frac{1-\tan \mathrm{A}}{1-\cot \mathrm{A}}\right)^{2}=\tan ^{2} \mathrm{~A}\)

If \(\tan \mathrm{A}=\cot \mathrm{B}\), prove that \(\mathrm{A}+\mathrm{B}=90^{\circ}\).

Evaluate: (i) \(\frac{\sin 18^{\circ}}{\cos 72^{\circ}}\) (ii) \(\frac{\tan 26^{\circ}}{\cot 64^{\circ}}\) (iii) \(\cos 48^{\circ}-\sin 42^{\circ}\) (iv) \(\operatorname{cosec} 31^{\circ}-\sec 59^{\circ}\)

Evaluate: (i) \(\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}\) (ii) \(\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}\)
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