what is $\sin 45^\circ$?

a. $\frac{1}{\sqrt{2}}$

b. $\frac{1}{2}$

c. 1

d. $\sqrt{2}$

Question

what is $\sin 45^\circ$?

a. $\frac{1}{\sqrt{2}}$

b. $\frac{1}{2}$

c. 1

d. $\sqrt{2}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Special Right Triangles", "45-45-90 Triangle Theorem" ], "new_concepts": [ "Trigonometric Ratios", "Sine Ratio" ], "current_concepts": [ "Special Right Triangles", "45-45-90 Triangle Theorem", "Trigonometric Ratios", "Sine Ratio" ] } ## Identify the given triangle properties Using the *Special Right Triangles* and *45-45-90 Triangle Theorem* knowledge points $$ \begin{aligned} &\text{Triangle Type: Right Isosceles } (45^\circ\text{-}45^\circ\text{-}90^\circ)\ &\text{Opposite side to } 45^\circ = 1\ &\text{Adjacent side to } 45^\circ = 1\ &\text{Hypotenuse} = \sqrt{2} \end{aligned} $$ ## Apply the definition of the sine ratio We define the sine of an angle in a right triangle as the ratio of the length of the opposite side to the length of the hypotenuse. $$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$ ## Calculate the value of sine 45 degrees Substitute the known side lengths from our triangle into the ratio. $$ \sin(45^\circ) = \frac{1}{\sqrt{2}} $$

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

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

(C) 1

(D) $\sqrt{2}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given triangle properties

Apply the definition of the sine ratio

Calculate the value of sine 45 degrees

Answer: