Question

sketch an angle θ in standard position such that θ has the least possible positive measure, and the point (0,4) is on the terminal side of θ. then find the values of the six trigonometric functions for the angle. rationalize denominators if applicable. do not use a calculator. choose the correct graph below. ○ a. graph, ○ b. graph, ○ c. graph

Explanation:

Part 1: Choosing the Correct Graph

An angle in standard position has its initial side on the positive x - axis. The terminal side passes through the point \((0,4)\), which is on the positive y - axis.

• Option A: The terminal side is on the positive x - axis, so this is not correct.
• Option B: The terminal side is on the positive y - axis (since the point \((0,4)\) is on the positive y - axis and the angle is formed by rotating from the positive x - axis to the positive y - axis), so this is the correct graph.
• Option C: The terminal side is on the negative y - axis, so this is not correct.

Part 2: Finding the Six Trigonometric Functions

For a point \((x,y)\) on the terminal side of an angle \(\theta\) in standard position, we define \(r=\sqrt{x^{2}+y^{2}}\), and the six trigonometric functions are:

• \(\sin\theta=\frac{y}{r}\)
• \(\cos\theta=\frac{x}{r}\)
• \(\tan\theta=\frac{y}{x}\) (undefined when \(x = 0\))
• \(\csc\theta=\frac{r}{y}\) (undefined when \(y = 0\))
• \(\sec\theta=\frac{r}{x}\) (undefined when \(x = 0\))
• \(\cot\theta=\frac{x}{y}\) (undefined when \(y = 0\))

Step 1: Identify \(x\), \(y\), and calculate \(r\)

Given the point \((x,y)=(0,4)\).
We calculate \(r\) using the formula \(r = \sqrt{x^{2}+y^{2}}\). Substitute \(x = 0\) and \(y = 4\) into the formula:
\(r=\sqrt{0^{2}+4^{2}}=\sqrt{0 + 16}=\sqrt{16}=4\)

Step 2: Calculate \(\sin\theta\)

Using the formula \(\sin\theta=\frac{y}{r}\), substitute \(y = 4\) and \(r = 4\):
\(\sin\theta=\frac{4}{4}=1\)

Step 3: Calculate \(\cos\theta\)

Using the formula \(\cos\theta=\frac{x}{r}\), substitute \(x = 0\) and \(r = 4\):
\(\cos\theta=\frac{0}{4}=0\)

Step 4: Calculate \(\tan\theta\)

Using the formula \(\tan\theta=\frac{y}{x}\), substitute \(x = 0\) and \(y = 4\). Since \(x = 0\), \(\tan\theta\) is undefined.

Step 5: Calculate \(\csc\theta\)

Using the formula \(\csc\theta=\frac{r}{y}\), substitute \(y = 4\) and \(r = 4\):
\(\csc\theta=\frac{4}{4}=1\)

Step 6: Calculate \(\sec\theta\)

Using the formula \(\sec\theta=\frac{r}{x}\), substitute \(x = 0\) and \(r = 4\). Since \(x = 0\), \(\sec\theta\) is undefined.

Step 7: Calculate \(\cot\theta\)

Using the formula \(\cot\theta=\frac{x}{y}\), substitute \(x = 0\) and \(y = 4\):
\(\cot\theta=\frac{0}{4}=0\)

Final Answers

• Correct graph: B
• \(\sin\theta=\boldsymbol{1}\)
• \(\cos\theta=\boldsymbol{0}\)
• \(\tan\theta\): undefined
• \(\csc\theta=\boldsymbol{1}\)
• \(\sec\theta\): undefined
• \(\cot\theta=\boldsymbol{0}\)

Answer:

An angle in standard position has its initial side on the positive x - axis. The terminal side passes through the point \((0,4)\), which is on the positive y - axis.

• Option A: The terminal side is on the positive x - axis, so this is not correct.
• Option B: The terminal side is on the positive y - axis (since the point \((0,4)\) is on the positive y - axis and the angle is formed by rotating from the positive x - axis to the positive y - axis), so this is the correct graph.
• Option C: The terminal side is on the negative y - axis, so this is not correct.

Part 2: Finding the Six Trigonometric Functions

For a point \((x,y)\) on the terminal side of an angle \(\theta\) in standard position, we define \(r=\sqrt{x^{2}+y^{2}}\), and the six trigonometric functions are:

• \(\sin\theta=\frac{y}{r}\)
• \(\cos\theta=\frac{x}{r}\)
• \(\tan\theta=\frac{y}{x}\) (undefined when \(x = 0\))
• \(\csc\theta=\frac{r}{y}\) (undefined when \(y = 0\))
• \(\sec\theta=\frac{r}{x}\) (undefined when \(x = 0\))
• \(\cot\theta=\frac{x}{y}\) (undefined when \(y = 0\))

Step 1: Identify \(x\), \(y\), and calculate \(r\)

Given the point \((x,y)=(0,4)\).
We calculate \(r\) using the formula \(r = \sqrt{x^{2}+y^{2}}\). Substitute \(x = 0\) and \(y = 4\) into the formula:
\(r=\sqrt{0^{2}+4^{2}}=\sqrt{0 + 16}=\sqrt{16}=4\)

Step 2: Calculate \(\sin\theta\)

Using the formula \(\sin\theta=\frac{y}{r}\), substitute \(y = 4\) and \(r = 4\):
\(\sin\theta=\frac{4}{4}=1\)

Step 3: Calculate \(\cos\theta\)

Using the formula \(\cos\theta=\frac{x}{r}\), substitute \(x = 0\) and \(r = 4\):
\(\cos\theta=\frac{0}{4}=0\)

Step 4: Calculate \(\tan\theta\)

Using the formula \(\tan\theta=\frac{y}{x}\), substitute \(x = 0\) and \(y = 4\). Since \(x = 0\), \(\tan\theta\) is undefined.

Step 5: Calculate \(\csc\theta\)

Using the formula \(\csc\theta=\frac{r}{y}\), substitute \(y = 4\) and \(r = 4\):
\(\csc\theta=\frac{4}{4}=1\)

Step 6: Calculate \(\sec\theta\)

Using the formula \(\sec\theta=\frac{r}{x}\), substitute \(x = 0\) and \(r = 4\). Since \(x = 0\), \(\sec\theta\) is undefined.

Step 7: Calculate \(\cot\theta\)

Using the formula \(\cot\theta=\frac{x}{y}\), substitute \(x = 0\) and \(y = 4\):
\(\cot\theta=\frac{0}{4}=0\)

Final Answers

• Correct graph: B
• \(\sin\theta=\boldsymbol{1}\)
• \(\cos\theta=\boldsymbol{0}\)
• \(\tan\theta\): undefined
• \(\csc\theta=\boldsymbol{1}\)
• \(\sec\theta\): undefined
• \(\cot\theta=\boldsymbol{0}\)