Find the values of the sine, cosine, and tangent for ZA C A 36ft B
24ft

Find the values of the sine, cosine, and tangent for ∠A

a. sin A = ,  cos A = ,  tan A =

b. sin A = ,  cos A = ,  tan A =

c. sin A = ,  cos A = ,  tan A =

d. sin A = ,  cos A = ,  tan A =

d. sin A = ,  cos A = ,  tan A =

The triangle for the question has been attached to this response.

As shown in the triangle;

AC = 36ft

BC = 24ft

ACB = 90°

To calculate the values of the sine, cosine, and tangent of ∠A;

i. First calculate the value of the missing side AB.

Using Pythagoras' theorem;

⇒ (AB)² = (AC)² + (BC)²

Substitute the values of AC and BC

⇒ (AB)² = (36)² + (24)²

Solve for AB

⇒ (AB)² = 1296 + 576

⇒ (AB)² = 1872

⇒ AB =

⇒ AB = ft

From the values of the sides, it can be noted that the side AB is the hypotenuse of the triangle since that is the longest side with a value of ft (43.27ft).

ii. Calculate the sine of ∠A (i.e sin A)

The sine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the hypotenuse side of the triangle. i.e

sin Ф =             -------------(i)

In this case,

Ф = A

opposite = 24ft (This is the opposite side to angle A)

hypotenuse = ft (This is the longest side of the triangle)

Substitute these values into equation (i) as follows;

sin A =

sin A =

Rationalize the result by multiplying both the numerator and denominator by

sin A =

sin A =

iii. Calculate the cosine of ∠A (i.e cos A)

The cosine of an angle (Ф) in a triangle is given by the ratio of the adjacent side to that angle to the hypotenuse side of the triangle. i.e

cos Ф =             -------------(ii)

In this case,

Ф = A

adjacent = 36ft (This is the adjecent side to angle A)

hypotenuse = ft (This is the longest side of the triangle)

Substitute these values into equation (ii) as follows;

cos A =

cos A =

Rationalize the result by multiplying both the numerator and denominator by

cos A =

cos A =

iii. Calculate the tangent of ∠A (i.e tan A)

The cosine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the adjacent side of the triangle. i.e

tan Ф =             -------------(iii)

In this case,

Ф = A

opposite = 24 ft (This is the opposite side to angle A)

adjacent = 36 ft (This is the adjacent side to angle A)

Substitute these values into equation (iii) as follows;

tan A =

tan A =

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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∠cbe and ∠deb are same-side interior angles.

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