Question

given that the point (80, -84) is on the terminal side of an angle, θ, find the exact value of the following:

sin(θ)=

cos(θ)=

tan(θ)=

csc(θ)=

sec(θ)=

cot(θ)=

question help: worked example 1

Explanation:

Step1: Calculate the radius \(r\)

Use the formula \(r=\sqrt{x^{2}+y^{2}}\), where \(x = 80\) and \(y=-84\).

$$
LATEXBLOCK0
$$

Step2: Calculate \(\sin(\theta)\)

Use the formula \(\sin(\theta)=\frac{y}{r}\), where \(y=-84\) and \(r = 116\).

$$ \sin(\theta)=\frac{-84}{116}=-\frac{21}{29} $$

Step3: Calculate \(\cos(\theta)\)

Use the formula \(\cos(\theta)=\frac{x}{r}\), where \(x = 80\) and \(r = 116\).

$$ \cos(\theta)=\frac{80}{116}=\frac{20}{29} $$

Step4: Calculate \(\tan(\theta)\)

Use the formula \(\tan(\theta)=\frac{y}{x}\), where \(x = 80\) and \(y=-84\).

$$ \tan(\theta)=\frac{-84}{80}=-\frac{21}{20} $$

Step5: Calculate \(\csc(\theta)\)

Use the formula \(\csc(\theta)=\frac{r}{y}\), where \(y=-84\) and \(r = 116\).

$$ \csc(\theta)=\frac{116}{-84}=-\frac{29}{21} $$

Step6: Calculate \(\sec(\theta)\)

Use the formula \(\sec(\theta)=\frac{r}{x}\), where \(x = 80\) and \(r = 116\).

$$ \sec(\theta)=\frac{116}{80}=\frac{29}{20} $$

Step7: Calculate \(\cot(\theta)\)

Use the formula \(\cot(\theta)=\frac{x}{y}\), where \(x = 80\) and \(y=-84\).

$$ \cot(\theta)=\frac{80}{-84}=-\frac{20}{21} $$

Answer:

\(\sin(\theta)=-\frac{21}{29}\)

\(\cos(\theta)=\frac{20}{29}\)

\(\tan(\theta)=-\frac{21}{20}\)

\(\csc(\theta)=-\frac{29}{21}\)

\(\sec(\theta)=\frac{29}{20}\)

\(\cot(\theta)=-\frac{20}{21}\)