Question

sketch an angle θ in standard position such that θ has the least possible positive measure and the point (6,8) is on the terminal side of θ. then find the exact values of the six trigonometric functions for θ. choose the correct graph below. ○ a. graph, ○ b. graph, ○ c. graph

Explanation:

Part 1: Choosing the Correct Graph

To determine the correct graph, we analyze the coordinates of the point \((6, 8)\). In the coordinate plane, \(x = 6\) (positive) and \(y = 8\) (positive), so the point lies in the first quadrant.

• Option A: The terminal side is in the second quadrant (negative \(x\), positive \(y\)) – incorrect.
• Option B: The terminal side is in the third quadrant (negative \(x\), negative \(y\)) – incorrect.
• Option C: The terminal side is in the first quadrant (positive \(x\), positive \(y\)) – 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 first find \(r\) (the distance from the origin to the point) using the formula \(r=\sqrt{x^{2}+y^{2}}\). Then we use the definitions of the trigonometric functions:

• \(\sin\theta=\frac{y}{r}\)
• \(\cos\theta=\frac{x}{r}\)
• \(\tan\theta=\frac{y}{x}\) (for \(x

eq0\))

• \(\csc\theta=\frac{r}{y}\) (for \(y

eq0\))

• \(\sec\theta=\frac{r}{x}\) (for \(x

eq0\))

• \(\cot\theta=\frac{x}{y}\) (for \(y

eq0\))

Step 1: Calculate \(r\)

Given \(x = 6\) and \(y = 8\), we calculate \(r\):
\[
r=\sqrt{x^{2}+y^{2}}=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10
\]

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

Using \(\sin\theta=\frac{y}{r}\):
\[
\sin\theta=\frac{8}{10}=\frac{4}{5}
\]

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

Using \(\cos\theta=\frac{x}{r}\):
\[
\cos\theta=\frac{6}{10}=\frac{3}{5}
\]

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

Using \(\tan\theta=\frac{y}{x}\):
\[
\tan\theta=\frac{8}{6}=\frac{4}{3}
\]

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

Using \(\csc\theta=\frac{r}{y}\):
\[
\csc\theta=\frac{10}{8}=\frac{5}{4}
\]

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

Using \(\sec\theta=\frac{r}{x}\):
\[
\sec\theta=\frac{10}{6}=\frac{5}{3}
\]

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

Using \(\cot\theta=\frac{x}{y}\):
\[
\cot\theta=\frac{6}{8}=\frac{3}{4}
\]

Answer:

(Graph):
C