Finding sin A + cos A from tan A
We are given that $\tan A = \frac{8}{15}$.
Recall that in a right-angled triangle, $\tan A = \frac{\text{opposite}}{\text{adjacent}}$.
Let the side opposite to angle A be $8k$ and the side adjacent to angle A be $15k$, where $k$ is a positive constant.
Calculating the Hypotenuse
Using the Pythagorean theorem, $hypotenuse^2 = opposite^2 + adjacent^2$:
$ \text{hypotenuse}^2 = (8k)^2 + (15k)^2 $
$ \text{hypotenuse}^2 = 64k^2 + 225k^2 $
$ \text{hypotenuse}^2 = 289k^2 $
$ \text{hypotenuse} = \sqrt{289k^2} = 17k $
Calculating sin A and cos A
Now, we can find $\sin A$ and $\cos A$:
- $\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8k}{17k} = \frac{8}{17}$
- $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15k}{17k} = \frac{15}{17}$
Calculating sin A + cos A
Finally, we calculate the sum:
$ \sin A + \cos A = \frac{8}{17} + \frac{15}{17} $
$ \sin A + \cos A = \frac{8+15}{17} $
$ \sin A + \cos A = \frac{23}{17} $
Thus, the value of $\sin A + \cos A$ is $\frac{23}{17}$.
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:
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- 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:
- Simplify: $\frac{\sin A \tan A}{(1 - \cos A)}$
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 ?
