Question
Find the value of other five trigonometric function: $\cos x = - \frac{1}{2}$, x lies in third quadrant.
✓
Answer
Here $\cos x = - \frac{1}{2}$
Now ${\sin ^2}x + {\cos ^2}x = 1 \Rightarrow {\sin ^2}x + {\left( { - \frac{1}{2}} \right)^2} = 1$$ \Rightarrow {\sin ^2}x = 1 - \frac{1}{4}$
$ \Rightarrow {\sin ^2}x = \frac{3}{4} \Rightarrow \sin x = \pm \frac{{\sqrt 3 }}{2}$
But x lies in third quadrant.
$\therefore \sin x = - \frac{{\sqrt 3 }}{2}$
Now cosec x $ = \frac{1}{{\sin x}} = - \frac{2}{{\sqrt 3 }}$ and $\sec x = \frac{1}{{\cos x}} = - 2$
$\tan x = \frac{{\sin x}}{{\cos }} = \frac{{ - \sqrt 3 /2}}{{ - 1/2}} = \sqrt 3 $ and $\cot x = \frac{{\cos x}}{{\sin x}} = \frac{{ - 1/2}}{{ - \sqrt 3 /2}} = \frac{1}{{\sqrt 3 }}$
Now ${\sin ^2}x + {\cos ^2}x = 1 \Rightarrow {\sin ^2}x + {\left( { - \frac{1}{2}} \right)^2} = 1$$ \Rightarrow {\sin ^2}x = 1 - \frac{1}{4}$
$ \Rightarrow {\sin ^2}x = \frac{3}{4} \Rightarrow \sin x = \pm \frac{{\sqrt 3 }}{2}$
But x lies in third quadrant.
$\therefore \sin x = - \frac{{\sqrt 3 }}{2}$
Now cosec x $ = \frac{1}{{\sin x}} = - \frac{2}{{\sqrt 3 }}$ and $\sec x = \frac{1}{{\cos x}} = - 2$
$\tan x = \frac{{\sin x}}{{\cos }} = \frac{{ - \sqrt 3 /2}}{{ - 1/2}} = \sqrt 3 $ and $\cot x = \frac{{\cos x}}{{\sin x}} = \frac{{ - 1/2}}{{ - \sqrt 3 /2}} = \frac{1}{{\sqrt 3 }}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the derivative of function $f(x) = \frac{{\cos x}}{{1 + \sin x}}$ (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers).
→If $\text{P}= \big\{\text{x} : \text{x} < 3, \text{x}\in\text{N}\big\},$ $\text{Q}= \big\{\text{x} : \text{x} \leq 2, \text{x}\in\text{W}\big\}.$ Find $(\text{P} \cup \text{Q}) \times (\text{P} \cap \text{Q}),$ where W is the set of whole numbers.
→Find the equation of the circle with centre $\left( \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$ and radius $\frac { 1 } { 12 }$.
→By giving a counter example, show that the following statements are not true.
→- $p:$ If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.
- $q:$ The equation $x^2– 1 = 0$ does not have a root lying between $0$ and $2.$
If $f: R \rightarrow R , f(x)=x^2$ then find-
(i) Range of $f$
(ii) $\{x / f(x)=4\}$
(iii) $\{y / f(y)=-1\}$
→(i) Range of $f$
(ii) $\{x / f(x)=4\}$
(iii) $\{y / f(y)=-1\}$
Consider the following population and year graph: Find the slope of the line AB and using it, find what will be the population in the year 2010.
Sketch the graphs of the following function:
$\text{g(x)}=3\sin\Big(\text{x}-\frac{\pi}{4}\Big),0\leq\text{x}\leq\frac{5\pi}{4}$
→$\text{g(x)}=3\sin\Big(\text{x}-\frac{\pi}{4}\Big),0\leq\text{x}\leq\frac{5\pi}{4}$
A sequence is defined by $\text{a}_\text{n}=\text{n}^3-6\text{n}^2-11\text{n}-6,\text{n}\in\text{N.}$ show that the first three terms of the sequence are zero and all other terms are positive.
→A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of numbers that turn up is 3.
Find the value of the following trigonomentric ratios:
$\sin\frac{5\pi}{3}$
→$\sin\frac{5\pi}{3}$