Question

sketch an angle θ in standard position such that θ has the least possible positive measure, and the point (-7,0) 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

To determine the correct graph for the angle \(\theta\) with the terminal side passing through \((-7, 0)\):

• A point \((x, y) = (-7, 0)\) lies on the negative \(x\)-axis (since \(y = 0\) and \(x<0\)).
• In standard position, an angle with its terminal side on the negative \(x\)-axis has a measure of \(180^\circ\) (or \(\pi\) radians) for the least positive measure.
• Analyzing the options:
• Option A: The terminal side is on the negative \(x\)-axis (matches \((-7, 0)\)).
• Option B: Terminal side is on the negative \(y\)-axis (incorrect, as \(y

eq0\) here).

• Option C: Terminal side is on the positive \(y\)-axis (incorrect, as \(x

eq0\) and \(x\) is not negative here).

So the correct graph is Option A.

Part 2: Finding the Six Trigonometric Functions

For a point \((x, y)\) on the terminal side of an angle \(\theta\) in standard position, we first find \(r=\sqrt{x^2 + y^2}\), where \(x=-7\), \(y = 0\).

Step 1: Calculate \(r\)

\(r=\sqrt{(-7)^2+0^2}=\sqrt{49 + 0}=7\)

Step 2: Sine Function (\(\sin\theta\))

\(\sin\theta=\frac{y}{r}=\frac{0}{7}=0\)

Step 3: Cosine Function (\(\cos\theta\))

\(\cos\theta=\frac{x}{r}=\frac{-7}{7}=-1\)

Step 4: Tangent Function (\(\tan\theta\))

\(\tan\theta=\frac{y}{x}=\frac{0}{-7}=0\) (Note: \(x
eq0\) here, so it's defined)

Step 5: Cosecant Function (\(\csc\theta\))

\(\csc\theta=\frac{r}{y}\), but \(y = 0\), so \(\csc\theta\) is undefined (division by zero).

Step 6: Secant Function (\(\sec\theta\))

\(\sec\theta=\frac{r}{x}=\frac{7}{-7}=-1\)

Step 7: Cotangent Function (\(\cot\theta\))

\(\cot\theta=\frac{x}{y}\), but \(y = 0\), so \(\cot\theta\) is undefined (division by zero).

Final Answers:

• Correct graph: \(\boldsymbol{\text{A}}\)
• Trigonometric functions:

\(\sin\theta = 0\), \(\cos\theta=-1\), \(\tan\theta = 0\), \(\csc\theta\) (undefined), \(\sec\theta=-1\), \(\cot\theta\) (undefined)

Answer:

To determine the correct graph for the angle \(\theta\) with the terminal side passing through \((-7, 0)\):

• A point \((x, y) = (-7, 0)\) lies on the negative \(x\)-axis (since \(y = 0\) and \(x<0\)).
• In standard position, an angle with its terminal side on the negative \(x\)-axis has a measure of \(180^\circ\) (or \(\pi\) radians) for the least positive measure.
• Analyzing the options:
• Option A: The terminal side is on the negative \(x\)-axis (matches \((-7, 0)\)).
• Option B: Terminal side is on the negative \(y\)-axis (incorrect, as \(y

eq0\) here).

• Option C: Terminal side is on the positive \(y\)-axis (incorrect, as \(x

eq0\) and \(x\) is not negative here).

So the correct graph is Option A.

Part 2: Finding the Six Trigonometric Functions

For a point \((x, y)\) on the terminal side of an angle \(\theta\) in standard position, we first find \(r=\sqrt{x^2 + y^2}\), where \(x=-7\), \(y = 0\).

Step 1: Calculate \(r\)

\(r=\sqrt{(-7)^2+0^2}=\sqrt{49 + 0}=7\)

Step 2: Sine Function (\(\sin\theta\))

\(\sin\theta=\frac{y}{r}=\frac{0}{7}=0\)

Step 3: Cosine Function (\(\cos\theta\))

\(\cos\theta=\frac{x}{r}=\frac{-7}{7}=-1\)

Step 4: Tangent Function (\(\tan\theta\))

\(\tan\theta=\frac{y}{x}=\frac{0}{-7}=0\) (Note: \(x
eq0\) here, so it's defined)

Step 5: Cosecant Function (\(\csc\theta\))

\(\csc\theta=\frac{r}{y}\), but \(y = 0\), so \(\csc\theta\) is undefined (division by zero).

Step 6: Secant Function (\(\sec\theta\))

\(\sec\theta=\frac{r}{x}=\frac{7}{-7}=-1\)

Step 7: Cotangent Function (\(\cot\theta\))

\(\cot\theta=\frac{x}{y}\), but \(y = 0\), so \(\cot\theta\) is undefined (division by zero).

Final Answers:

• Correct graph: \(\boldsymbol{\text{A}}\)
• Trigonometric functions:

\(\sin\theta = 0\), \(\cos\theta=-1\), \(\tan\theta = 0\), \(\csc\theta\) (undefined), \(\sec\theta=-1\), \(\cot\theta\) (undefined)