Question

example: determine trig function values given a determine trig function values given a point on the terminal side of an angle given that the point (24, -7) 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 post to forum submit question

Explanation:

Step1: Calculate the radius r

For a point $(x,y)=(24, - 7)$ on the terminal - side of an angle, use the formula $r=\sqrt{x^{2}+y^{2}}$. So, $r=\sqrt{24^{2}+(-7)^{2}}=\sqrt{576 + 49}=\sqrt{625}=25$.

Step2: Calculate $\sin(\theta)$

By the definition $\sin(\theta)=\frac{y}{r}$. Substituting $y=-7$ and $r = 25$, we get $\sin(\theta)=-\frac{7}{25}$.

Step3: Calculate $\cos(\theta)$

By the definition $\cos(\theta)=\frac{x}{r}$. Substituting $x = 24$ and $r = 25$, we get $\cos(\theta)=\frac{24}{25}$.

Step4: Calculate $\tan(\theta)$

By the definition $\tan(\theta)=\frac{y}{x}$. Substituting $x = 24$ and $y=-7$, we get $\tan(\theta)=-\frac{7}{24}$.

Step5: Calculate $\csc(\theta)$

Since $\csc(\theta)=\frac{1}{\sin(\theta)}$, substituting $\sin(\theta)=-\frac{7}{25}$, we get $\csc(\theta)=-\frac{25}{7}$.

Step6: Calculate $\sec(\theta)$

Since $\sec(\theta)=\frac{1}{\cos(\theta)}$, substituting $\cos(\theta)=\frac{24}{25}$, we get $\sec(\theta)=\frac{25}{24}$.

Step7: Calculate $\cot(\theta)$

Since $\cot(\theta)=\frac{1}{\tan(\theta)}$, substituting $\tan(\theta)=-\frac{7}{24}$, we get $\cot(\theta)=-\frac{24}{7}$.

Answer:

$\sin(\theta)=-\frac{7}{25}$
$\cos(\theta)=\frac{24}{25}$
$\tan(\theta)=-\frac{7}{24}$
$\csc(\theta)=-\frac{25}{7}$
$\sec(\theta)=\frac{25}{24}$
$\cot(\theta)=-\frac{24}{7}$