Question

What is the value of $c$?

Answer 1

Hint: We first describe the Pythagorean identity with respect to the right-angle triangle. We use the formula of ${\left(base\right)}^2+{\left(height\right)}^2={\left(hypotenuse\right)}^2$. Putting the values, we get ${10}^2+{24}^2=c^2$. We solve to find the value of $c$.

Complete step by step solution:
The Pythagorean identity is about the trigonometric identity that is used in case of right-angle triangles.
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
We express it for $\mathrm{\Delta }ABC$ where $\mathrm{\angle }B={\displaystyle \frac{\pi }{2}}$.

We have the expression for the triangle as ${\left(base\right)}^2+{\left(height\right)}^2={\left(hypotenuse\right)}^2$.
For $\mathrm{\Delta }ABC$, $base=BC=b,height=AB=a,hypotenuse=AC=c$
Applying the rule, we get ${\left(a\right)}^2+{\left(b\right)}^2={\left(c\right)}^2$.
We can also express it with respect to the angle $\mathrm{\angle }BCA$. Let $\mathrm{\angle }BCA=\theta$.
The equation ${\left(a\right)}^2+{\left(b\right)}^2={\left(c\right)}^2$ can be simplified
$\begin{array}{rl}& a^2+b^2=c^2\\ & \Rightarrow {\left({\displaystyle \frac{a}{c}}\right)}^2+{\left({\displaystyle \frac{b}{c}}\right)}^2=1\end{array}$
With respect to the angle $\mathrm{\angle }BCA=\theta$, we have ${\displaystyle \frac{a}{c}}=\sin \theta ,{\displaystyle \frac{b}{c}}=\cos \theta$.
Replacing the values, we get ${\sin }^2\theta +{\cos }^2\theta =1$.
There are many reformed versions of the formula ${\sin }^2\theta +{\cos }^2\theta =1$.
Dividing with ${\cos }^2\theta$, we got $${\tan }^2\theta +1={\sec }^2\theta$$.
Dividing with ${\sin }^2\theta$, we got $${\cot }^2\theta +1={\csc }^2\theta$$.
For the given image we have 10 as height and 24 as base.
Therefore, $c^2={10}^2+{24}^2=100+576=676$.
Taking square root, we get $c=\sqrt{676}=26$.
We have the length of the hypotenuse as 26 units.

Note: The concept of Pythagorean identity is similar for both angles and sides. The representation of the triangle has to be for the right-angle triangle.
The figure shows how the sign of the sine function varies as the angle changes quadrant.