# The angle theta is in the fourth quadrant and costheta=frac{2}{7}. Find the exact value of the remaining five trigonometric functions.

Question
Trigonometry
The angle $$\displaystyle\theta$$ is in the fourth quadrant and $$\displaystyle{\cos{\theta}}={\frac{{{2}}}{{{7}}}}$$. Find the exact value of the remaining five trigonometric functions.

2021-02-10
It is given that $$\displaystyle\theta$$ is in the fourth quadrant and
$$\displaystyle{\cos{{\left(\theta\right)}}}={\frac{{{2}}}{{{7}}}}$$
Since,
$$\displaystyle{\cos{{\left(\theta\right)}}}={\frac{{\text{Base(B)}}}{{\text{Hypotenuse(H)}}}}{\cos{\theta}}$$ and $$\displaystyle{\sec{\theta}}$$
$$\displaystyle{\frac{{{B}}}{{{H}}}}$$=$$\displaystyle{\frac{{{2}}}{{{7}}}}$$
Then,
B=2, H=7
Use Pythagoras formula to find the perpindicular P,
$$\displaystyle{H}^{{2}}={B}^{{2}}+{P}^{{2}}$$
$$\displaystyle{P}^{{2}}={H}^{{2}}-{B}^{{2}}$$
$$\displaystyle={7}^{{2}}-{2}^{{2}}$$
$$\displaystyle={49}-{4}$$
$$\displaystyle={45}$$
That is,
$$\displaystyle{P}=\sqrt{{{45}}}$$
All the trigonometric ratios except $$\displaystyle{\cos{\theta}}$$ and $$\displaystyle{\sec{\theta}}$$ are negative as $$\displaystyle\theta$$ is in the fourth quadrant. Then
$$\displaystyle{\sec{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}$$
$$\displaystyle={\frac{{{1}}}{{{\frac{{{2}}}{{{7}}}}}}}$$
$$\displaystyle={\frac{{{7}}}{{{2}}}}$$
And $$\displaystyle{\sin{\theta}}=-{\frac{{{P}}}{{{H}}}}=-{\frac{{\sqrt{{{45}}}}}{{{7}}}}$$
$$\displaystyle{\csc{\theta}}=-{\frac{{{H}}}{{{P}}}}=-{\frac{{{7}}}{{\sqrt{{{45}}}}}}$$
$$\displaystyle{\tan{\theta}}=-{\frac{{{P}}}{{{B}}}}=-{\frac{{\sqrt{{{45}}}}}{{{2}}}}$$
$$\displaystyle{\cot{\theta}}=-{\frac{{{B}}}{{{P}}}}=-{\frac{{{2}}}{{\sqrt{{{45}}}}}}$$
Hence, the required value of the remaining trigonometric ratios is
$$\displaystyle{\sec{\theta}}={\frac{{{7}}}{{{2}}}},{\sin{\theta}}=-{\frac{{\sqrt{{{45}}}}}{{{7}}}},{\csc{\theta}}=-{\frac{{{7}}}{{\sqrt{{{45}}}}}},{\tan{\theta}}=-{\frac{{\sqrt{{45}}}}{{{2}}}},{\cot{\theta}}=-{\frac{{{2}}}{{\sqrt{{{45}}}}}}$$

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