Question

the terminal side of an angle with measure θ in standard position passes through the given point. sketch the reference triangle. evaluate sin θ, cos θ, and tan θ. simplify all radicals.

1. p(5, 12)
2. p(-5, 12)
3. p(-5, -12)
4. p(5, -12)
5. p(-6, -8)
6. p(6, -8)
7. p(6, 8)
8. p(-6, 8)
9. p(10, -24)
10. p(5, 10)
11. p(-6, 4)
12. p(-12, -10)

trigonometric functions 15

Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{12}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{5}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{12}{5}\)
- # Answer:
\(\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{12}{5}\)
2. **For point \(P(- 5,12)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-5)^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{12}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-5}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{12}{-5}=-\frac{12}{5}\)
- # Answer:
\(\sin\theta=\frac{12}{13},\cos\theta=-\frac{5}{13},\tan\theta=-\frac{12}{5}\)
3. **For point \(P(-5,-12)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-5)^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-12}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-5}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-12}{-5}=\frac{12}{5}\)
- # Answer:
\(\sin\theta=-\frac{12}{13},\cos\theta=-\frac{5}{13},\tan\theta=\frac{12}{5}\)
4. **For point \(P(5,-12)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{5^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-12}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{5}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-12}{5}=-\frac{12}{5}\)
- # Answer:
\(\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}\)
5. **For point \(P(-6,-8)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-6)^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-8}{10}=-\frac{4}{5}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-6}{10}=-\frac{3}{5}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-8}{-6}=\frac{4}{3}\)
- # Answer:
\(\sin\theta=-\frac{4}{5},\cos\theta=-\frac{3}{5},\tan\theta=\frac{4}{3}\)
6. **For point \(P(6,-8)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{6^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-8}{10}=-\frac{4}{5}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-8}{6}=-\frac{4}{3}\)
- # Answer:
\(\sin\theta=-\frac{4}{5},\cos\theta=\frac{3}{5},\tan\theta=-\frac{4}{3}\)
7. **For point \(P(6,8)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{8}{10}=\frac{4}{5}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{8}{6}=\frac{4}{3}\)
- # Answer:
\(\sin\theta=\frac{4}{5},\cos\theta=\frac{3}{5},\tan\theta=\frac{4}{3}\)
8. **For point \(P(-6,8)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-6)^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{8}{10}=\frac{4}{5}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-6}{10}=-\frac{3}{5}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{8}{-6}=-\frac{4}{3}\)
- # Answer:
\(\sin\theta=\frac{4}{5},\cos\theta=-\frac{3}{5},\tan\theta=-\frac{4}{3}\)
9. **For point \(P(10,-24)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{10^{2}+(-24)^{2}}=\sqrt{100 + 576}=\sqrt{676}=26\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-24}{26}=-\frac{12}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{10}{26}=\frac{5}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-24}{10}=-\frac{12}{5}\)
- # Answer:
\(\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}\)
10. **For point \(P(5,10)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{5^{2}+10^{2}}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{10}{5\sqrt{5}}=\frac{2\sqrt{5}}{5}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{5}{5\sqrt{5}}=\frac{\sqrt{5}}{5}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{10}{5}=2\)
- # Answer:
\(\sin\theta=\frac{2\sqrt{5}}{5},\cos\theta=\frac{\sqrt{5}}{5},\tan\theta = 2\)
11. **For point \(P(-6,4)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-6)^{2}+4^{2}}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{4}{2\sqrt{13}}=\frac{2\sqrt{13}}{13}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-6}{2\sqrt{13}}=-\frac{3\sqrt{13}}{13}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{4}{-6}=-\frac{2}{3}\)
- # Answer:
\(\sin\theta=\frac{2\sqrt{13}}{13},\cos\theta=-\frac{3\sqrt{13}}{13},\tan\theta=-\frac{2}{3}\)
12. **For point \(P(-12,-10)\)**:
- # Explanation:
## Step1: Calculate the value of \(r\)
\(r=\sqrt{(-12)^{2}+(-10)^{2}}=\sqrt{144 + 100}=\sqrt{244}=2\sqrt{61}\)
## Step2: Calculate \(\sin\theta\)
\(\sin\theta=\frac{y}{r}=\frac{-10}{2\sqrt{61}}=-\frac{5\sqrt{61}}{61}\)
## Step3: Calculate \(\cos\theta\)
\(\cos\theta=\frac{x}{r}=\frac{-12}{2\sqrt{61}}=-\frac{6\sqrt{61}}{61}\)
## Step4: Calculate \(\tan\theta\)
\(\tan\theta=\frac{y}{x}=\frac{-10}{-12}=\frac{5}{6}\)
- # Answer:
\(\sin\theta=-\frac{5\sqrt{61}}{61},\cos\theta=-\frac{6\sqrt{61}}{61},\tan\theta=\frac{5}{6}\)

Answer:

1. For point \(P(5,12)\):

• First, find the distance \(r\) from the origin \((0,0)\) to the point \((x,y)=(5,12)\) using the formula \(r = \sqrt{x^{2}+y^{2}}\).
• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{12}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{5}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{12}{5}\)

• # Answer:

\(\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=\frac{12}{5}\)

1. For point \(P(- 5,12)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-5)^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{12}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-5}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{12}{-5}=-\frac{12}{5}\)

• # Answer:

\(\sin\theta=\frac{12}{13},\cos\theta=-\frac{5}{13},\tan\theta=-\frac{12}{5}\)

1. For point \(P(-5,-12)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-5)^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-12}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-5}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-12}{-5}=\frac{12}{5}\)

• # Answer:

\(\sin\theta=-\frac{12}{13},\cos\theta=-\frac{5}{13},\tan\theta=\frac{12}{5}\)

1. For point \(P(5,-12)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{5^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-12}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{5}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-12}{5}=-\frac{12}{5}\)

• # Answer:

\(\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}\)

1. For point \(P(-6,-8)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-6)^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-8}{10}=-\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-6}{10}=-\frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-8}{-6}=\frac{4}{3}\)

• # Answer:

\(\sin\theta=-\frac{4}{5},\cos\theta=-\frac{3}{5},\tan\theta=\frac{4}{3}\)

1. For point \(P(6,-8)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{6^{2}+(-8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-8}{10}=-\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-8}{6}=-\frac{4}{3}\)

• # Answer:

\(\sin\theta=-\frac{4}{5},\cos\theta=\frac{3}{5},\tan\theta=-\frac{4}{3}\)

1. For point \(P(6,8)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{8}{10}=\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{6}{10}=\frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{8}{6}=\frac{4}{3}\)

• # Answer:

\(\sin\theta=\frac{4}{5},\cos\theta=\frac{3}{5},\tan\theta=\frac{4}{3}\)

1. For point \(P(-6,8)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-6)^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{8}{10}=\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-6}{10}=-\frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{8}{-6}=-\frac{4}{3}\)

• # Answer:

\(\sin\theta=\frac{4}{5},\cos\theta=-\frac{3}{5},\tan\theta=-\frac{4}{3}\)

1. For point \(P(10,-24)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{10^{2}+(-24)^{2}}=\sqrt{100 + 576}=\sqrt{676}=26\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-24}{26}=-\frac{12}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{10}{26}=\frac{5}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-24}{10}=-\frac{12}{5}\)

• # Answer:

\(\sin\theta=-\frac{12}{13},\cos\theta=\frac{5}{13},\tan\theta=-\frac{12}{5}\)

1. For point \(P(5,10)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{5^{2}+10^{2}}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{10}{5\sqrt{5}}=\frac{2\sqrt{5}}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{5}{5\sqrt{5}}=\frac{\sqrt{5}}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{10}{5}=2\)

• # Answer:

\(\sin\theta=\frac{2\sqrt{5}}{5},\cos\theta=\frac{\sqrt{5}}{5},\tan\theta = 2\)

1. For point \(P(-6,4)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-6)^{2}+4^{2}}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{4}{2\sqrt{13}}=\frac{2\sqrt{13}}{13}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-6}{2\sqrt{13}}=-\frac{3\sqrt{13}}{13}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{4}{-6}=-\frac{2}{3}\)

• # Answer:

\(\sin\theta=\frac{2\sqrt{13}}{13},\cos\theta=-\frac{3\sqrt{13}}{13},\tan\theta=-\frac{2}{3}\)

1. For point \(P(-12,-10)\):

• # Explanation:

Step1: Calculate the value of \(r\)

\(r=\sqrt{(-12)^{2}+(-10)^{2}}=\sqrt{144 + 100}=\sqrt{244}=2\sqrt{61}\)

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}=\frac{-10}{2\sqrt{61}}=-\frac{5\sqrt{61}}{61}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}=\frac{-12}{2\sqrt{61}}=-\frac{6\sqrt{61}}{61}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}=\frac{-10}{-12}=\frac{5}{6}\)

• # Answer:

\(\sin\theta=-\frac{5\sqrt{61}}{61},\cos\theta=-\frac{6\sqrt{61}}{61},\tan\theta=\frac{5}{6}\)