all six trig functions using the unit circle: \four corners\

$\theta = 0^\circ / 360^\circ$
$\cos(0) = \frac{1}{1}$
$\sec(0) = \frac{1}{1}$
$\sin(0) = \frac{0}{1}$
$\csc(0) = \frac{1}{0}$
$\tan(0) = \frac{0}{1}$
$\cot(0) = \frac{1}{0}$
radians: $\theta = 0 / 2\pi$

$\theta = 180^\circ$
$\cos(180) = -1$
$\sec(180) =$
$\sin(180) = 0$
$\csc(180) =$
$\tan(180) = 0$
$\cot(180) =$
radian: $\theta = \pi$

$\theta = 90^\circ$
$\cos(180) = \frac{0}{1}$
$\sec(180) =$
$\sin(180) = \frac{1}{1}$
$\csc(180) =$
$\tan(180) = und$
$\cot(180) = \frac{0}{1}$
radian: $\theta = \frac{\pi}{2}$

$\theta = 270^\circ$
$\cos(270) = 0$
$\sec(270) =$
$\sin(270) = -1$
$\csc(270) =$
$\tan(270) = und$
$\cot(270) =$
radian: $\theta = \frac{3\pi}{2}$

Question

all six trig functions using the unit circle: \four corners\

$\theta = 0^\circ / 360^\circ$
$\cos(0) = \frac{1}{1}$
$\sec(0) = \frac{1}{1}$
$\sin(0) = \frac{0}{1}$
$\csc(0) = \frac{1}{0}$
$\tan(0) = \frac{0}{1}$
$\cot(0) = \frac{1}{0}$
radians: $\theta = 0 / 2\pi$

$\theta = 180^\circ$
$\cos(180) = -1$
$\sec(180) =$
$\sin(180) = 0$
$\csc(180) =$
$\tan(180) = 0$
$\cot(180) =$
radian: $\theta = \pi$

$\theta = 90^\circ$
$\cos(180) = \frac{0}{1}$
$\sec(180) =$
$\sin(180) = \frac{1}{1}$
$\csc(180) =$
$\tan(180) = und$
$\cot(180) = \frac{0}{1}$
radian: $\theta = \frac{\pi}{2}$

$\theta = 270^\circ$
$\cos(270) = 0$
$\sec(270) =$
$\sin(270) = -1$
$\csc(270) =$
$\tan(270) = und$
$\cot(270) =$
radian: $\theta = \frac{3\pi}{2}$

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Ratios", "Reciprocal Trigonometric Functions" ], "new_concepts": [ "Quadrantal Angles", "Unit Circle Coordinates" ], "current_concepts": [ "Trigonometric Ratios", "Reciprocal Trigonometric Functions", "Quadrantal Angles", "Unit Circle Coordinates" ] } ## Evaluate functions at 0 degrees Using the *Trigonometric Ratios* and *Reciprocal Trigonometric Functions* knowledge points $$ \begin{aligned} &\cos(0) = \frac{1}{1} = 1\ &\sin(0) = \frac{0}{1} = 0\ &\tan(0) = \frac{0}{1} = 0\ &\sec(0) = \frac{1}{1} = 1\ &\csc(0) = \frac{1}{0} \implies \text{undefined}\ &\cot(0) = \frac{1}{0} \implies \text{undefined} \end{aligned} $$ ## Evaluate functions at 180 degrees Using the *Trigonometric Ratios* and *Reciprocal Trigonometric Functions* knowledge points $$ \begin{aligned} &\cos(180^\circ) = -1\ &\sin(180^\circ) = 0\ &\tan(180^\circ) = 0\ &\sec(180^\circ) = \frac{1}{-1} = -1\ &\csc(180^\circ) = \frac{1}{0} \implies \text{undefined}\ &\cot(180^\circ) = \frac{1}{0} \implies \text{undefined} \end{aligned} $$ ## Evaluate functions at 90 degrees Using the *Trigonometric Ratios* and *Reciprocal Trigonometric Functions* knowledge points $$ \begin{aligned} &\cos(90^\circ) = \frac{0}{1} = 0\ &\sin(90^\circ) = \frac{1}{1} = 1\ &\tan(90^\circ) = \text{undefined}\ &\sec(90^\circ) = \frac{1}{0} \implies \text{undefined}\ &\csc(90^\circ) = \frac{1}{1} = 1\ &\cot(90^\circ) = \frac{0}{1} = 0 \end{aligned} $$ ## Evaluate functions at 270 degrees Using the *Trigonometric Ratios* and *Reciprocal Trigonometric Functions* knowledge points $$ \begin{aligned} &\cos(270^\circ) = 0\ &\sin(270^\circ) = -1\ &\tan(270^\circ) = \text{undefined}\ &\sec(270^\circ) = \frac{1}{0} \implies \text{undefined}\ &\csc(270^\circ) = \frac{1}{-1} = -1\ &\cot(270^\circ) = \frac{0}{1} = 0 \end{aligned} $$ ## Question 1: $\theta = 0^\circ / 360^\circ$ (Radians: $\theta = 0 / 2\pi$) * $\cos(0) = 1$ * $\sin(0) = 0$ * $\tan(0) = 0$ * $\sec(0) = 1$ * $\csc(0) = \text{undefined}$ * $\cot(0) = \text{undefined}$ ## Question 2: $\theta = 180^\circ$ (Radian: $\theta = \pi$) * $\cos(180^\circ) = -1$ * $\sin(180^\circ) = 0$ * $\tan(180^\circ) = 0$ * $\sec(180^\circ) = -1$ * $\csc(180^\circ) = \text{undefined}$ * $\cot(180^\circ) = \text{undefined}$ ## Question 3: $\theta = 90^\circ$ (Radian: $\theta = \frac{\pi}{2}$) * $\cos(90^\circ) = 0$ * $\sin(90^\circ) = 1$ * $\tan(90^\circ) = \text{undefined}$ * $\sec(90^\circ) = \text{undefined}$ * $\csc(90^\circ) = 1$ * $\cot(90^\circ) = 0$ ## Question 4: $\theta = 270^\circ$ (Radian: $\theta = \frac{3\pi}{2}$) * $\cos(270^\circ) = 0$ * $\sin(270^\circ) = -1$ * $\tan(270^\circ) = \text{undefined}$ * $\sec(270^\circ) = \text{undefined}$ * $\csc(270^\circ) = -1$ * $\cot(270^\circ) = 0$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Quadrantal Angles" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate functions at 0 degrees

Evaluate functions at 180 degrees

Evaluate functions at 90 degrees

Evaluate functions at 270 degrees

Answer:

Question 1: \(\theta = 0^\circ / 360^\circ\) (Radians: \(\theta = 0 / 2\pi\))

Question 2: \(\theta = 180^\circ\) (Radian: \(\theta = \pi\))

Question 3: \(\theta = 90^\circ\) (Radian: \(\theta = \frac{\pi}{2}\))

Question 4: \(\theta = 270^\circ\) (Radian: \(\theta = \frac{3\pi}{2}\))