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Chapter 2: Problem 75

Solve for \(\boldsymbol{x}\) : $$ 5 \tan ^{-1} x+3 \cot ^{-1} x=\frac{7 \pi}{4} $$

Short Answer

Expert verified
The solution to the equation is: \( x = \tan( \frac{\frac{7\pi}{4} - \frac{3\pi}{2}}{2}) \)

Step by step solution

01

Apply the Relationship

The given equation is \(5 \tan^{-1}x + 3 \cot^{-1}x = \frac{7\pi}{4} \). Recall the relationship between \(\tan^{-1}x\) and \(\cot^{-1}x\), which can be written as \(\cot^{-1}x = \frac{\pi}{2} - \tan^{-1}x\). Substitute \(\cot^{-1}x\) by \(\frac{\pi}{2} - \tan^{-1}x\) in the given equation to get: \(5 \tan^{-1}x + 3[\frac{\pi}{2} - \tan^{-1}x] = \frac{7\pi}{4}\)
02

Simplify the Equation

After substitution in step 1, simplify the equation by multiplying through the brackets and combining like terms: \(5 \tan^{-1}x + \frac{3\pi}{2} - 3 \tan^{-1}x= \frac{7\pi}{4} \). Rearrange the equation to isolate terms involving x on one side to get: \(2 \tan^{-1}x = \frac{7\pi}{4} - \frac{3\pi}{2} \).
03

Solve for x

After arranging the equation in step 2, divide through by 2 to solve for \(\tan^{-1}x\): \(\tan^{-1}x = \frac{\frac{7\pi}{4} - \frac{3\pi}{2}}{2}\). Then use the inverse tangent function (take the tangent of both sides) to solve for x: \(x = \tan( \frac{\frac{7\pi}{4} - \frac{3\pi}{2}}{2} ) \)

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Key Concepts

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

tangent inverse
When we talk about the tangent inverse, or \( an^{-1}(x)\), we refer to the angle whose tangent is \(x\). This angle is usually represented within the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The tangent inverse is a fundamental concept in trigonometry, as it lets us find angles when the tangent is known.
  • To use \(\tan^{-1}(x)\), you need to understand that it is essentially the opposite function of the tangent.
  • The function \(\tan^{-1}(x)\) helps in solving equations where the tangent values are known and you need to find the original angle.
  • In the context of the exercise, we solve for \(x\) by isolating \(\tan^{-1}(x)\) and then applying the tangent function on both sides of the equation.
Using this concept, we can transform trigonometric problems into simple algebraic ones, allowing us to find solutions more easily.
cotangent inverse
The cotangent inverse, often noted as \(\cot^{-1}(x)\), is the angle whose cotangent is \(x\). It is related to the tangent inverse because \(\cot^{-1}(x)\) can be rewritten using the identity \(\cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x)\). This identity is incredibly useful in solving problems where both functions appear.
  • In our equation, this identity allows us to substitute \(\cot^{-1}(x)\) and simplify the equation into terms of \(\tan^{-1}(x)\) only.
  • By simplifying equations using \(\cot^{-1}(x)\), we can often reduce the complexity and make trigonometric solutions easier to compute.
This transformation helps us transition from working with two variables to a single unified term, streamlining the solution process.
angle relationships
Understanding angle relationships is key in trigonometric solutions. Inverse trigonometric functions often rely on these relationships to solve equations effectively.
  • Angle relationships reveal how different angles interact. For example, the complementary angle relationship shows that \(\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}\). This is a crucial identity because it reduces the burden of solving equations with multiple angles.
  • These relationships simplify complex expressions, and are derived from the properties of circles and triangles.
  • Knowing how these angles relate helps us make substitutions and solve for unknowns faster, as seen in the exercise.
By leveraging these relationships, we can solve problems with more efficiency and fewer computational steps.

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Most popular questions from this chapter

Find the values of: $$ \cos ^{-1}(\sin (-5))+\sin ^{-1}(\cos (-5)) $$

Prove that: $$ \tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\right) $$

Prove that: If \(\tan ^{-1} x+\tan ^{-1} y=\frac{\pi}{4}\), then prove that \(x+y+x y=1\).

Solve for \(\boldsymbol{x}\) : $$ \sin ^{-1}(x)+\sin ^{-1}(3 x)=\frac{\pi}{3} $$

Find the values of: $$ \sin ^{-1}(\sin 50)+\cos ^{-1}(\cos 50)-\tan ^{-1}(\tan 50) $$
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