Problem 8
Question
Find the value of \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tan ^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right)\)
Step-by-Step Solution
Verified Answer
The value of \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tan ^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right)\) simplifies to \(\tan^{-1}\frac{1}{3}\).
1Step 1: Apply the property of tan inverse
Given is \(\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)-\tan ^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right)\). We can apply the property \(\tan ^{-1}a - tan^{-1}b = \tan ^{-1}(\frac{a - b}{1 + ab})\) here.
2Step 2: Substitute the values
Substitute the values of a and b into the equation, where a equals \(\frac{1}{\sqrt{2}}\) and b equals \(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\). So it becomes \(\tan ^{-1}(\frac{\frac{1}{\sqrt{2}} - \frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}}{1 + \frac{1}{\sqrt{2}} * \frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}}\).
3Step 3: Simplify the equation
Simplify the equation now. The final simplified equation is \(\tan^{-1}\frac{1}{3}\).
4Step 4: Calculate the value of \(\tan^{-1}\frac{1}{3}\)
Calculate the value of \(\tan^{-1}\frac{1}{3}\) using basic trigonometric knowledge or calculator.
Key Concepts
Understanding Tan Inverse FormulasSimplifying Trigonometric ExpressionsNavigating Trigonometric Equations
Understanding Tan Inverse Formulas
Inverse trigonometric functions allow us to find an angle when the value of a trigonometric ratio is known. One such function is the inverse tangent or arctangent, often denoted as \( \tan^{-1} \), which is a reversal of the tangent function. The \( \tan^{-1} \) of a value 'x' finds the angle whose tangent is 'x'.
There are key formulas to ease the manipulation of expressions involving \( \tan^{-1} \). One such formula is:\[\tan ^{-1}a - \tan^{-1}b = \tan ^{-1}\left(\frac{a - b}{1 + ab}\right)\]This formula is extremely useful when we are tasked with simplifying an expression containing the difference of two \( \tan^{-1} \) terms. It allows us to combine these terms into a single \( \tan^{-1} \) term, thus simplifying the calculation.
There are key formulas to ease the manipulation of expressions involving \( \tan^{-1} \). One such formula is:\[\tan ^{-1}a - \tan^{-1}b = \tan ^{-1}\left(\frac{a - b}{1 + ab}\right)\]This formula is extremely useful when we are tasked with simplifying an expression containing the difference of two \( \tan^{-1} \) terms. It allows us to combine these terms into a single \( \tan^{-1} \) term, thus simplifying the calculation.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is a vital skill in mathematics, enabling students to solve trigonometric equations efficiently. Simplification often involves applying identities, properties, and formulas.
For example, to make sense of complex expressions like \( \tan^{-1}\left(\frac{1}{\sqrt{2}}\right) - \tan^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right) \), we use the aforementioned subtraction formula for \( \tan^{-1} \) terms.
It's essential to combine terms by aligning similar elements, such as rationalizing denominators, combining like terms, and factoring where appropriate. The goal is to rewrite the expression in a way that is more direct and easier to interpret or compute. The process often ends with a function that accepts an angle value or an algebraic number as input, thus streamlining the path to the final solution.
For example, to make sense of complex expressions like \( \tan^{-1}\left(\frac{1}{\sqrt{2}}\right) - \tan^{-1}\left(\frac{\sqrt{5-2 \sqrt{6}}}{1+\sqrt{6}}\right) \), we use the aforementioned subtraction formula for \( \tan^{-1} \) terms.
It's essential to combine terms by aligning similar elements, such as rationalizing denominators, combining like terms, and factoring where appropriate. The goal is to rewrite the expression in a way that is more direct and easier to interpret or compute. The process often ends with a function that accepts an angle value or an algebraic number as input, thus streamlining the path to the final solution.
Navigating Trigonometric Equations
Trigonometric equations involve terms that include trigonometric functions such as sine, cosine, and tangent. Solving these equations often requires one to find all the angles that satisfy the equation.
For example, after simplifying expressions, we might reach a point where we need to calculate \( \tan^{-1}\frac{1}{3} \). Here, we are essentially equating the tangent of an unknown angle to \( \frac{1}{3} \), forming a basic trigonometric equation. The solution involves finding the angle whose tangent gives \( \frac{1}{3} \).It's important to remember that, due to the periodic nature of trigonometric functions, there might be multiple angles that satisfy the equation. We determine the principal value in the range \( -\frac{\pi}{2} < y < \frac{\pi}{2} \) for \( \tan^{-1} \) but remember other angles might be relevant depending on the context. Problem-solving also involves using specific tools like calculators or unit circles to find these values accurately.
For example, after simplifying expressions, we might reach a point where we need to calculate \( \tan^{-1}\frac{1}{3} \). Here, we are essentially equating the tangent of an unknown angle to \( \frac{1}{3} \), forming a basic trigonometric equation. The solution involves finding the angle whose tangent gives \( \frac{1}{3} \).It's important to remember that, due to the periodic nature of trigonometric functions, there might be multiple angles that satisfy the equation. We determine the principal value in the range \( -\frac{\pi}{2} < y < \frac{\pi}{2} \) for \( \tan^{-1} \) but remember other angles might be relevant depending on the context. Problem-solving also involves using specific tools like calculators or unit circles to find these values accurately.
Other exercises in this chapter
Problem 7
Find the simplest value of \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right), \forall x \in\left(\frac{1}{2}, 1\right)\)
View solution Problem 7
The number of solutions of the equation \(\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}\) (a) 0 (b) 1 (c) 2 (d) More than two
View solution Problem 8
The smallest and the largest values of \(\tan ^{-1}\left(\frac{1-x}{1+x}\right), 0 \leq x \leq 1\) are (a) \(0, \pi\) (b) \(0, \frac{\pi}{4}\) (c) \(-\frac{\pi}
View solution Problem 9
Let \(m=\sin ^{-1}\left(a^{6}+1\right)+\cos ^{-1}\left(a^{4}+1\right)-\tan ^{-1}\left(a^{2}+1\right)\), then find the image of the line \(x+y=m\) about the \(y\
View solution