Question

If we have the trigonometric ratio as $8\tan A=15$, then find the value of the expression $\sin A-\cos A$.

Answer 1

Hint: Here, first draw the right-angled triangle and use the trigonometric ratios to find sin A and cos A from tan A. We will get all the values of the opposite side, adjacent side, and the hypotenuse. In the end, after finding sin A and cos A, subtract them according to the question and obtain the required result.

Complete step-by-step solution:
We have been given the question that, $8\tan A=15$. From this, we can write, $\tan A=\dfrac{15}{8}$.
Now, let us draw a right-angled triangle and indicate the angle A.

According to the trigonometric ratios, we know,
$\tan \theta =\dfrac{\text{Opposite}}{\text{Adjacent}}$
We have, $\tan A=\dfrac{15}{8}$.
Therefore, on comparing the above equations we can see that, the opposite side = 15 and the adjacent side = 8.
We also know that,
$\sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}}=\dfrac{\text{O}}{\text{H}}$
So, we have the opposite side = 15, so we can write,
$\sin A=\dfrac{15}{\text{H}}$
We also know that,
$\cos \theta =\dfrac{\text{Adjacent}}{\text{Hypotenuse}}=\dfrac{\text{A}}{\text{H}}$
We have the adjacent side = 8, so we can write,
$\cos A=\dfrac{8}{\text{H}}$
We know that Pythagoras theorem is given by,
${{\left( \text{Hypotenuse} \right)}^{\text{2}}}\text{=}{{\left( \text{Opposite side} \right)}^{\text{2}}}\text{+}{{\left( \text{Adjacent side} \right)}^{\text{2}}}$
On substituting the values, we get,
$\begin{align}
  & {{\left( \text{Hypotenuse} \right)}^{\text{2}}}\text{=}{{\left( 15 \right)}^{\text{2}}}\text{+}{{\left( 8 \right)}^{\text{2}}} \\
 & =225+64 \\
 & =289 \\
\end{align}$
On taking the square root on both the sides, we get,
\[\text{Hypotenuse =}\sqrt{289}=17\]
Now, on substituting the value of hypotenuse in $\sin A=\dfrac{15}{\text{H}}$ and $\cos A=\dfrac{8}{\text{H}}$, we get,
$\sin A=\dfrac{15}{17}$ and $\cos A=\dfrac{8}{17}$
Now, we are supposed to find $\sin A-\cos A$, so we will get,
$\begin{align}
  & \sin A-\cos A=\dfrac{15}{17}-\dfrac{8}{17} \\
 & =\dfrac{15-8}{17} \\
 & =\dfrac{7}{17} \\
\end{align}$
Therefore, we get the value of $\sin A-\cos A=\dfrac{7}{17}$.

Note: Trigonometry is the branch of mathematics that deals with the relations of sides and angles of triangles and with the relevant functions of any angles. There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant, and cotangent. The hypotenuse is known as the longest side in the right-angled triangle. We can only use the Pythagoras theorem, if there is an angle of ${{90}^{\circ }}$ in the triangle.
Out of all the six trigonometric functions, sine, cosine, and tangent are most commonly used to solve problems related to the trigonometric equations. In the first quadrant, all the trigonometric functions are positive, in the second quadrant, only sine and cosecant are positive, in the third quadrant, only tangent and cotangent are positive, and in the fourth quadrant, only cosine and secant are the positive trigonometric functions.