which of these is the correct interpretation of the ordered pair $(-1, 0)$ on the unit circle?

- the sine of $3\pi/2$ radians is $-1$ and the cosine of $3\pi/2$ radians is $0$.
- the sine of $\pi$ radians is $-1$ and the cosine of $\pi$ radians is $0$.
- the cosine of $2\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.
- the cosine of $\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.

Question

which of these is the correct interpretation of the ordered pair $(-1, 0)$ on the unit circle?

- the sine of $3\pi/2$ radians is $-1$ and the cosine of $3\pi/2$ radians is $0$.
- the sine of $\pi$ radians is $-1$ and the cosine of $\pi$ radians is $0$.
- the cosine of $2\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.
- the cosine of $\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Unit Circle Trigonometry" ], "new_concepts": [], "current_concepts": [ "Unit Circle Trigonometry", "Trigonometric Ratios" ] } ## Identify coordinates on the unit circle Using the *Unit Circle Trigonometry* knowledge point $$ \begin{aligned} &P(\theta) = (x, y) = (\cos\theta, \sin\theta)\ &(x, y) = (-1, 0) \implies \cos\theta = -1,\quad \sin\theta = 0 \end{aligned} $$ ## Determine the angle theta Using the *Unit Circle Trigonometry* knowledge point $$ \begin{aligned} &\theta = \pi \text{ radians}\ &\cos(\pi) = -1\ &\sin(\pi) = 0 \end{aligned} $$ ## Evaluate the given options Using the *Unit Circle Trigonometry* knowledge point $$ \begin{aligned} &\text{Option 1: } \sin(3\pi/2) = -1, \cos(3\pi/2) = 0 \quad (\text{Incorrect angle})\ &\text{Option 2: } \sin(\pi) = -1, \cos(\pi) = 0 \quad (\text{Values swapped})\ &\text{Option 3: } \cos(2\pi) = -1, \sin(\pi) = 0 \quad (\text{Incorrect } \cos(2\pi) = 1)\ &\text{Option 4: } \cos(\pi) = -1, \sin(\pi) = 0 \quad (\text{Correct}) \end{aligned} $$

The sine of $3\pi/2$ radians is $-1$ and the cosine of $3\pi/2$ radians is $0$.

The sine of $\pi$ radians is $-1$ and the cosine of $\pi$ radians is $0$.

The cosine of $2\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.

The cosine of $\pi$ radians is $-1$ and the sine of $\pi$ radians is $0$.

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QUESTION IMAGE

Question

Explanation:

Identify coordinates on the unit circle

Determine the angle theta

Evaluate the given options

Answer: