given right triangle xyz, what is the value of $\tan(60^\circ)$?

$\bigcirc \frac{1}{2}$
$\bigcirc \frac{\sqrt{3}}{2}$
$\bigcirc \sqrt{3}$
$\bigcirc \frac{2}{1}$

Question

given right triangle xyz, what is the value of $\tan(60^\circ)$?

$\bigcirc \frac{1}{2}$
$\bigcirc \frac{\sqrt{3}}{2}$
$\bigcirc \sqrt{3}$
$\bigcirc \frac{2}{1}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Right Triangle Trigonometry", "Trigonometric Ratios" ], "new_concepts": [], "current_concepts": [ "Right Triangle Trigonometry", "Trigonometric Ratios" ] } ## Identify the given values and target ratio $$ \begin{aligned} &\text{Angle } X = 60^\circ\ &\text{Adjacent side to } 60^\circ \text{ (side } XZ\text{)} = 21\ &\text{Hypotenuse (side } XY\text{)} = 42 \end{aligned} $$ ## Calculate the opposite side using the Pythagorean theorem $$ \begin{aligned} &YZ = \sqrt{XY^2 - XZ^2}\ &YZ = \sqrt{42^2 - 21^2} = \sqrt{1764 - 441} = \sqrt{1323} = 21\sqrt{3} \end{aligned} $$ ## Compute the tangent of 60 degrees $$ \begin{aligned} &\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{YZ}{XZ}\ &\tan(60^\circ) = \frac{21\sqrt{3}}{21} = \sqrt{3} \end{aligned} $$

(A) $\frac{1}{2}$

(B) $\frac{\sqrt{3}}{2}$

(C) $\sqrt{3}$

(D) $\frac{2}{1}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Right Triangle Trigonometry" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and target ratio

Calculate the opposite side using the Pythagorean theorem

Compute the tangent of 60 degrees

Answer: