sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios in a right triangle.
cosecant (csc), secant (sec), and cotangent (cot) are their reciprocal functions.
a great way to remember which functions match up is that each pair has a \co.\

all six trig functions from a right triangle:

example:
right triangle with sides 5, 12, 13 and angle $\theta$
$\sin\theta = \frac{12}{13}$, $\csc\theta = \frac{13}{12}$
$\cos\theta = \frac{5}{13}$, $\sec\theta = \frac{13}{5}$
$\tan\theta = \frac{12}{5}$, $\cot\theta = \frac{5}{12}$

you do #1:
right triangle with sides 3, 4, 5 and angle $\theta$

you do #2:
right triangle with sides 20, 21, 29 and angle $\theta$

another way to write the reciprocal functions:
$\sin\theta = \frac{1}{\csc\theta}$, $\csc\theta = \frac{1}{\sin\theta}$
$\cos\theta = \frac{1}{\sec\theta}$, $\sec\theta = \frac{1}{\cos\theta}$
$\tan\theta = \frac{1}{\cot\theta}$, $\cot\theta = \frac{1}{\tan\theta}$
$\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Question

sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios in a right triangle.
cosecant (csc), secant (sec), and cotangent (cot) are their reciprocal functions.
a great way to remember which functions match up is that each pair has a \co.\

all six trig functions from a right triangle:

example:
right triangle with sides 5, 12, 13 and angle $\theta$
$\sin\theta = \frac{12}{13}$, $\csc\theta = \frac{13}{12}$
$\cos\theta = \frac{5}{13}$, $\sec\theta = \frac{13}{5}$
$\tan\theta = \frac{12}{5}$, $\cot\theta = \frac{5}{12}$

you do #1:
right triangle with sides 3, 4, 5 and angle $\theta$

you do #2:
right triangle with sides 20, 21, 29 and angle $\theta$

another way to write the reciprocal functions:
$\sin\theta = \frac{1}{\csc\theta}$, $\csc\theta = \frac{1}{\sin\theta}$
$\cos\theta = \frac{1}{\sec\theta}$, $\sec\theta = \frac{1}{\cos\theta}$
$\tan\theta = \frac{1}{\cot\theta}$, $\cot\theta = \frac{1}{\tan\theta}$
$\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Ratios", "Reciprocal Trigonometric Functions", "Right Triangle Trigonometry" ], "new_concepts": [], "current_concepts": [ "Trigonometric Ratios", "Reciprocal Trigonometric Functions", "Right Triangle Trigonometry" ] } ## Identify sides for You Do #1 $$ \begin{aligned} &\text{Hypotenuse } (H) = 5\ &\text{Adjacent } (A) = 3\ &\text{Opposite } (O) = 4 \end{aligned} $$ ## Calculate ratios for You Do #1 $$ \begin{aligned} &\sin\theta = \frac{O}{H} = \frac{4}{5}, \quad \csc\theta = \frac{H}{O} = \frac{5}{4}\ &\cos\theta = \frac{A}{H} = \frac{3}{5}, \quad \sec\theta = \frac{H}{A} = \frac{5}{3}\ &\tan\theta = \frac{O}{A} = \frac{4}{3}, \quad \cot\theta = \frac{A}{O} = \frac{3}{4} \end{aligned} $$ ## Identify sides and calculate ratios for You Do #2 $$ \begin{aligned} &H = 29, \quad A = 20, \quad O = 21\ &\sin\theta = \frac{21}{29}, \quad \csc\theta = \frac{29}{21}\ &\cos\theta = \frac{20}{29}, \quad \sec\theta = \frac{29}{20}\ &\tan\theta = \frac{21}{20}, \quad \cot\theta = \frac{20}{21} \end{aligned} $$ ## Question 1: You Do #1 $$ \begin{aligned} \sin\theta &= \frac{4}{5} & \csc\theta &= \frac{5}{4} \ \cos\theta &= \frac{3}{5} & \sec\theta &= \frac{5}{3} \ \tan\theta &= \frac{4}{3} & \cot\theta &= \frac{3}{4} \end{aligned} $$ ## Question 2: You Do #2 $$ \begin{aligned} \sin\theta &= \frac{21}{29} & \csc\theta &= \frac{29}{21} \ \cos\theta &= \frac{20}{29} & \sec\theta &= \frac{29}{20} \ \tan\theta &= \frac{21}{20} & \cot\theta &= \frac{20}{21} \end{aligned} $$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Right Triangle Trigonometry" ] }

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QUESTION IMAGE

Question

Explanation:

Identify sides for You Do #1

Calculate ratios for You Do #1

Identify sides and calculate ratios for You Do #2

Answer:

Question 1: You Do #1

Question 2: You Do #2