Evaluate Trigonometric Expression
To find the value of the given trigonometric expression, we substitute the known values of trigonometric functions for standard angles and simplify.
Identify Standard Trigonometric Values
The standard values required for the calculation are:
- $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$
- $\sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \sqrt{2}$
- $\tan 45^{\circ} = 1$
- $\cos 60^{\circ} = \frac{1}{2}$
- $\sin 30^{\circ} = \frac{1}{2}$
- $\sin 90^{\circ} = 1$
Calculate Expression Value
The expression is: $\sin 45^{\circ} \sec 45^{\circ} \tan 45^{\circ} + \cos 60^{\circ} \sin 30^{\circ} \sin 90^{\circ} - 4\cos 60^{\circ}$
Substitute the values into the expression:
$ \left( \frac{1}{\sqrt{2}} \times \sqrt{2} \times 1 \right) + \left( \frac{1}{2} \times \frac{1}{2} \times 1 \right) - \left( 4 \times \frac{1}{2} \right) $
Simplify each part of the expression:
- Part 1: $\frac{1}{\sqrt{2}} \times \sqrt{2} \times 1 = 1$
- Part 2: $\frac{1}{2} \times \frac{1}{2} \times 1 = \frac{1}{4}$
- Part 3: $4 \times \frac{1}{2} = 2$
Combine the simplified parts according to the original expression:
$ 1 + \frac{1}{4} - 2 $
Perform the final arithmetic operations:
$ \frac{4}{4} + \frac{1}{4} - \frac{8}{4} = \frac{4 + 1 - 8}{4} = \frac{5 - 8}{4} = \frac{-3}{4} $
The value of the expression is $\frac{-3}{4}$.
Similar Questions
- If $\cot A = \sqrt{5}$, then find the value of $\frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A}$
- Find the value of $\frac{\cos^6 \theta + \sin^6 \theta + 3 \sin^2 \theta \cos^2 \theta}{\csc \theta \sec \theta ((\sin \theta + \cos \theta)^2 - 1)}$
- The value of $2\sin30^{\circ} + 5\cos60^{\circ} + 3\sin180^{\circ}$ is:
If $\tan\theta = \frac{5}{12}$, $0 < \theta < \frac{\pi}{2}$, then the value of $\frac{\cos\theta + 5\cot\theta}{\text{cosec}\theta - \cos\theta}$ will be:
- If $\cos^4\theta - \sin^4\theta = \frac{2}{3}$ then the value of $1 - 2\sin^2\theta$ is:
- If $\sin(3A - 20^\circ) = \cos(20^\circ - 3B)$, then the value of A - B is:
- If $\tan A = \frac{8}{15}$ then find $\sin A + \cos A$.
- If $\sin \theta - \cos \theta = 0$, and $0 \le \theta \le 90^\circ$, then the value of $\sin \theta + \cos \theta$ is:
- If $1 + \tan \theta = \sqrt{3}$, then $\sqrt{3} \cot \theta - 1 = ?$
- $\cos(x - y) = \frac{\sqrt{3}}{2}$, and $\sin(x + y) = 1$, where $x$ and $y$ are positive acute angles and $x \ge y$, then $x$ and $y$ are:
Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
