What is the value of cos (2 cos -1 0.8)?
0.28
Understanding the Question: Evaluating a Trigonometric Expression
The question asks us to find the value of the expression cos (2 cos-1 0.8). This involves understanding inverse trigonometric functions and trigonometric identities, specifically the double angle identity for cosine.
Step-by-Step Calculation of cos (2 cos⁻¹ 0.8)
Let's break down the expression to make it easier to solve.
- Define the inverse cosine term: Let $\theta = \cos^{-1} (0.8)$. This means that the cosine of the angle $\theta$ is 0.8. So, $\cos \theta = 0.8$.
- Rewrite the original expression: The original expression is $\cos (2 \cos^{-1} 0.8)$. Substituting $\theta = \cos^{-1} 0.8$, the expression becomes $\cos (2\theta)$.
- Apply the double angle identity for cosine: We need to find the value of $\cos (2\theta)$. One of the double angle identities for cosine is: $\cos (2\theta) = 2\cos^2 \theta - 1$
- Substitute the value of cos θ: We know that $\cos \theta = 0.8$. Substitute this value into the identity: $\cos (2\theta) = 2 \times (0.8)^2 - 1$
- Perform the calculation: First, calculate the square of 0.8: $(0.8)^2 = 0.8 \times 0.8 = 0.64$ Now, substitute this back into the expression: $\cos (2\theta) = 2 \times 0.64 - 1$ Multiply 2 by 0.64: $2 \times 0.64 = 1.28$ Finally, subtract 1: $1.28 - 1 = 0.28$
Thus, the value of $\cos (2 \cos^{-1} 0.8)$ is 0.28.
Revision Table: Key Trigonometric Concepts
| Concept | Definition/Formula |
|---|---|
| Inverse Cosine ($\cos^{-1} x$ or $\arccos x$) | The angle $\theta$ such that $\cos \theta = x$. The principal value range is usually $[0, \pi]$. |
| Double Angle Identity for Cosine | $\cos (2\theta) = \cos^2 \theta - \sin^2 \theta$ $\cos (2\theta) = 2\cos^2 \theta - 1$ $\cos (2\theta) = 1 - 2\sin^2 \theta$ |
Additional Information on Inverse Trigonometric Functions
Inverse trigonometric functions, also known as arc functions, do the opposite of the standard trigonometric functions. For example, $\cos^{-1}(x)$ gives you the angle whose cosine is $x$. It's important to remember the domain and range of these functions.
- For $\cos^{-1} x$:
- Domain: $[-1, 1]$ (because the value of cosine is always between -1 and 1)
- Range (Principal Value): $[0, \pi]$ (0 to 180 degrees)
In this problem, 0.8 is within the domain $[-1, 1]$, so $\cos^{-1} (0.8)$ is a valid angle within the range $[0, \pi]$. The double angle $2\theta$ might fall outside this range, but the cosine of any angle is always defined.
Similar Questions
- If \(4 \sin^{-1} x + \cos^{-1} x = \pi\), then what is \(\sin^{-1} x + 4 \cos^{-1} x\) equal to?
- What is \(\cot^2 (\sec^{-1}2) + \tan^2 (\text{cosec}^{-1}3)\) equal to?
- Consider the following in respect of inverse circular functions :
I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ? - What is \(\tan \left[ 2 \tan^{-1} \frac{1}{2} - \frac{\pi}{4} \right]\) equal to ?
For what value of θ, where 0 < θ < \(\frac{\pi}{2}\) , does sin θ + sin θ cos θ maximum value?
What is the principal value of \({\sin ^{ - 1}}\left( {\sin \frac{{2{\rm{\pi }}}}{3}} \right)?\) ?
- \(\rm tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\) holds, when
Consider the following statements:
1. \({\tan ^{ - 1}}{\rm{x}} + {\tan ^{ - 1}}\left( {\frac{1}{{\rm{x}}}} \right) = {\rm{\pi }}\)
2. There exist x, y ∈ [-1, 1], where x ≠ y such that sin -1 x + cos -1 \({\rm{y}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?The value of \({\rm{tan}}\left( {2{{\tan }^{ - 1}}\frac{1}{5} - \frac{\pi }{4}} \right)\) is
Consider the following:
1. \({\sin ^{ - 1}}\frac{4}{5} + {\sin ^{ - 1}}\frac{3}{5} = \frac{\pi }{2}\)
2. \({\tan ^{ - 1}}\sqrt 3 + {\tan ^{ - 1}}1 = - {\tan ^{ - 1}}\left( {2 + \sqrt 3 } \right)\)
Which of the above is/are correct?
Important Questions from Inverse Trigonometric Functions
Solve for x?
sin 2x - 4 sin x + 3 = 0, 0 ≤ x ≤ \(π/2\)
What is the principal solutions of the equation \(\tan x=-\frac{1}{\sqrt{3}}\) ?
- If \(4 \sin^{-1} x + \cos^{-1} x = \pi\), then what is \(\sin^{-1} x + 4 \cos^{-1} x\) equal to?
- What is \(\cot^2 (\sec^{-1}2) + \tan^2 (\text{cosec}^{-1}3)\) equal to?
- Consider the following in respect of inverse circular functions :
I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ?
