Question

If cosθ=−2/3 and tanθ<0, thensin(θ)=_________;tan(θ)=_________ ;cot(θ)=_________;sec(θ)=_________;csc(θ)=_________;Give exact values.

Answer 1

Answer::

`\begin{gathered} \sin (\theta)=\frac{\sqrt[]{5}}{3},\textcolor{red}{tan(\theta)=-\frac{\sqrt[]{5}}{2},}\cot (\theta)=-\frac{2\sqrt[]{5}}{5} \\ \sec (\theta)=-(3)/(2),\textcolor{red}{\csc (\theta)=\frac{3\sqrt[]{5}}{5}} \end{gathered}`

Explanation:

Given:

`\begin{gathered} \cos \theta=-(2)/(3) \\ \tan \theta<0 \end{gathered}`

If the cosine and tangent of an angle are both negative, then the angle is in Quadrant II.

First, determine the length of the opposite side using the Pythagorean Theorem.

`\begin{gathered} \cos \theta=(-2)/(3)\implies\text{Adjacent}=-2,\; \text{Hypotenuse}=3 \\ \text{Hyp}^2=\text{Adj}^2+\text{Opp}^2 \\ 3^2=(-2)^2+\text{Opp}^2 \\ \text{Opp}^2=9-4=5 \\ \text{Opposite}=√(5) \end{gathered}`

The length of the opposite side is √5.

Therefore:

`\begin{gathered} \sin (\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{\sqrt[]{5}}{3} \\ \tan (\theta)=\frac{\text{Opposite}}{\text{Adjacent}}=-\frac{\sqrt[]{5}}{2} \end{gathered}`

Cotangent is the inverse of tangent, therefore:

`\begin{gathered} \cot (\theta)=(1)/(\tan(\theta))=-\frac{2}{\sqrt[]{5}} \\ \text{Rationalise the denominator} \\ =-\frac{2}{\sqrt[]{5}}*\frac{\sqrt[]{5}}{\sqrt[]{5}} \\ \implies\cot (\theta)=-\frac{2\sqrt[]{5}}{5} \end{gathered}`

Secant is the inverse of Cosine, therefore:

`\sec (\theta)=(1)/(\cos(\theta))=-(3)/(2)`

Cosecant is the inverse of Sine, therefore:

`\begin{gathered} \csc (\theta)=(1)/(\sin(\theta))=\frac{3}{\sqrt[]{5}} \\ \text{Rationalise the denominator} \\ =\frac{3}{\sqrt[]{5}}*\frac{\sqrt[]{5}}{\sqrt[]{5}} \\ \implies\csc (\theta)=\frac{3\sqrt[]{5}}{5} \end{gathered}`