# The two legs of a right triangle are $\sin \theta +\sin \left( \dfrac{3\pi }{2}-\theta \right)$ and $\cos \theta -\cos \left( \dfrac{3\pi }{2}-\theta \right)$. The length of its hypotenuse is:

A. $1$

B. $\sqrt{2}$

C. $2$

D. Some function of $\theta $

Answer

Verified

361.2k+ views

Hint: The given problem is related to the value of sine and cosine of an angle in the third quadrant. Sine and cosine function are negative in the third quadrant. Use this property to find the lengths of the legs of the right triangle. Then use the Pythagoras theorem to determine the length of its hypotenuse.

Complete step-by-step answer:

We know, any angle in the third quadrant is of the form $\left( \dfrac{3\pi }{2}-\theta \right)$ . We also know that sine and cosine functions are negative in the third quadrant. So, the value of $\sin \left( \dfrac{3\pi }{2}-\theta \right)$ will be $-\cos \theta $ and the value of $\cos \left( \dfrac{3\pi }{2}-\theta \right)$ will be $-\sin \theta $ . Now, the length of the legs of the right triangle are given as $\sin \theta +\sin \left( \dfrac{3\pi }{2}-\theta \right)$ and $\cos \theta -\cos \left( \dfrac{3\pi }{2}-\theta \right)$ . But we know that the value of $\sin \left( \dfrac{3\pi }{2}-\theta \right)$ is $-\cos \theta $ and the value of $\cos \left( \dfrac{3\pi }{2}-\theta \right)$ is $-\sin \theta $ . So, the length of the legs of the right triangle are $\sin \theta -\cos \theta $ and $\cos \theta +\sin \theta $ .

Now, we need to find the length of its hypotenuse. We know, the Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the perpendicular sides. So, if $a,b$ and $c$ are the lengths of sides of a right triangle such that $c>b,a$ , then according to the Pythagoras theorem, ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$ .

Now, in the given right triangle, the length of the legs are $\sin \theta -\cos \theta $ and $\cos \theta +\sin \theta $ . Let $h$ be the length of the hypotenuse. So, according to the Pythagoras theorem, \[{{h}^{2}}={{\left( \sin \theta -\cos \theta \right)}^{2}}+{{\left( \cos \theta +\sin \theta \right)}^{2}}\].

$\Rightarrow {{h}^{2}}={{\sin }^{2}}\theta +{{\cos }^{2}}\theta -2\sin \theta \cos \theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta +2\sin \theta \cos \theta $

Now, we know ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ . So, ${{h}^{2}}=1+1=2$ .

$\Rightarrow h=\sqrt{2}$

Hence, the length of the hypotenuse is $\sqrt{2}$ . Hence, option B. is the correct option.

Note: Some students get confused and write ${{\sin }^{2}}\theta -{{\cos }^{2}}\theta =1$ instead of ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Such mistakes should be avoided as it can result in getting wrong answers.

Complete step-by-step answer:

We know, any angle in the third quadrant is of the form $\left( \dfrac{3\pi }{2}-\theta \right)$ . We also know that sine and cosine functions are negative in the third quadrant. So, the value of $\sin \left( \dfrac{3\pi }{2}-\theta \right)$ will be $-\cos \theta $ and the value of $\cos \left( \dfrac{3\pi }{2}-\theta \right)$ will be $-\sin \theta $ . Now, the length of the legs of the right triangle are given as $\sin \theta +\sin \left( \dfrac{3\pi }{2}-\theta \right)$ and $\cos \theta -\cos \left( \dfrac{3\pi }{2}-\theta \right)$ . But we know that the value of $\sin \left( \dfrac{3\pi }{2}-\theta \right)$ is $-\cos \theta $ and the value of $\cos \left( \dfrac{3\pi }{2}-\theta \right)$ is $-\sin \theta $ . So, the length of the legs of the right triangle are $\sin \theta -\cos \theta $ and $\cos \theta +\sin \theta $ .

Now, we need to find the length of its hypotenuse. We know, the Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the perpendicular sides. So, if $a,b$ and $c$ are the lengths of sides of a right triangle such that $c>b,a$ , then according to the Pythagoras theorem, ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$ .

Now, in the given right triangle, the length of the legs are $\sin \theta -\cos \theta $ and $\cos \theta +\sin \theta $ . Let $h$ be the length of the hypotenuse. So, according to the Pythagoras theorem, \[{{h}^{2}}={{\left( \sin \theta -\cos \theta \right)}^{2}}+{{\left( \cos \theta +\sin \theta \right)}^{2}}\].

$\Rightarrow {{h}^{2}}={{\sin }^{2}}\theta +{{\cos }^{2}}\theta -2\sin \theta \cos \theta +{{\cos }^{2}}\theta +{{\sin }^{2}}\theta +2\sin \theta \cos \theta $

Now, we know ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ . So, ${{h}^{2}}=1+1=2$ .

$\Rightarrow h=\sqrt{2}$

Hence, the length of the hypotenuse is $\sqrt{2}$ . Hence, option B. is the correct option.

Note: Some students get confused and write ${{\sin }^{2}}\theta -{{\cos }^{2}}\theta =1$ instead of ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. Such mistakes should be avoided as it can result in getting wrong answers.

Last updated date: 25th Sep 2023

â€¢

Total views: 361.2k

â€¢

Views today: 11.61k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE