Triangle ABC is a right triangle and sin(53o) = StartFraction 4 Over x EndFraction. Solve for x and round to the nearest whole number.

Triangle A B C is shown. Angle A C B is a right angle and angle B A C is 53 degrees. The length of B C is 4 centimeters, the length of A C is y, and the length of hypotenuse A B is x.

Which equation correctly uses the value of x to represent the cosine of angle A?

cos(53o) = StartFraction 4 Over x EndFraction
cos(53o) = StartFraction y Over 5 EndFraction
cos(53o) = StartFraction x Over 4 EndFraction
cos(53o) = StartFraction 5 Over y EndFraction

The missing value of x is evaluated to be 5 cm approx. The equation correctly using the value of x to represent the cosine of angle A is: Option B: [tex]\cos(53^\circ) = \dfrac{y}{5}[/tex]

What are the six trigonometric ratios?

Trigonometric ratios for a right angled triangle are from the perspective of a particular non-right angle.

In a right angled triangle, two such angles are there which are not right angled(not of 90 degrees).

The slant side is called hypotenuse.

From the considered angle, the side opposite to it is called perpendicular, and the remaining side will be called base.

From that angle (suppose its measure is θ),

[tex]\sin(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of Hypotenuse}}\\\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}\\\\\tan(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of base}}\\\\\cot(\theta) = \dfrac{\text{Length of base}}{\text{Length of perpendicular}}\\\\\sec(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of base}}\\\\\csc(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of perpendicular}}\\[/tex]

For this case, we are given that:

Angle ACB is right angled.
Angle BAC is of 53 degrees
Length of BC = |BC| = 4 cm.
Length of AC = |AC| = y cm
Length of hypotenuse = |AB| = x cm

Referring to the figure attached below, we get:

[tex]sin(53^\circ) = \dfrac{|BC|}{|AB|} = \dfrac{4}{x}[/tex]

From calculator, we get:

[tex]sin(53^\circ) \approx 0.7986\\\dfrac{4}{x} \approx 0.7986\\\\x \approx \dfrac{4}{0.7986} \approx 5 \: \rm cm[/tex]

Thus, we get:

[tex]\cos(53^\circ) = \cos{\angle BAC = \dfrac{|AC|}{|AB|} = \dfrac{y}{x} = \dfrac{y}{5}[/tex]

Thus, the equation correctly using the value of x to represent the cosine of angle A is: Option B: [tex]\cos(53^\circ) = \dfrac{y}{5}[/tex]

Learn more about trigonometric ratios here:

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