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<h2>Question:</h2>
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;
<em>i. First calculate the value of the missing side AB.</em>
<em>Using Pythagoras' theorem;</em>
⇒ (AB)² = (AC)² + (BC)²
<em>Substitute the values of AC and BC</em>
⇒ (AB)² = (36)² + (24)²
<em>Solve for AB</em>
⇒ (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).
<em>ii. Calculate the sine of ∠A (i.e sin A)</em>
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)
<em>In this case,</em>
Ф = A
opposite = 24ft (This is the opposite side to angle A)
hypotenuse = ft (This is the longest side of the triangle)
<em>Substitute these values into equation (i) as follows;</em>
sin A =
sin A =
<em>Rationalize the result by multiplying both the numerator and denominator by </em><em />
sin A =
sin A =
<em>iii. Calculate the cosine of ∠A (i.e cos A)</em>
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)
<em>In this case,</em>
Ф = A
adjacent = 36ft (This is the adjecent side to angle A)
hypotenuse = ft (This is the longest side of the triangle)
<em>Substitute these values into equation (ii) as follows;</em>
cos A =
cos A =
<em>Rationalize the result by multiplying both the numerator and denominator by </em><em />
cos A =
cos A =
<em>iii. Calculate the tangent of ∠A (i.e tan A)</em>
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)
<em>In this case,</em>
Ф = A
opposite = 24 ft (This is the opposite side to angle A)
adjacent = 36 ft (This is the adjacent side to angle A)
<em>Substitute these values into equation (iii) as follows;</em>
tan A =
tan A =