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UNIT SEVEN
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
, 2
Graph: A triangle with vertices labelled A, B, and C. The side opposite angle A has a length of 8. The side opposite
angle B has a length of 6. The angle at vertex C is the right angle, indicated by a small square. The angle at vertex
A is labelled as α, and the angle at vertex B is labelled as θ.
1. Calculate tan 2θ
From the given triangle, we can identify the sides relative to angle θ (angle at B). The side opposite θ is the vertical
side, which has a length of 6. The side adjacent to θ is the horizontal side, which has a length of 8. The hypotenuse
is the remaining side, which we can calculate using the Pythagorean theorem but is not needed for the tangent ratio.
Therefore, tan θ = opposite/adjacent = 6/8 = 3/4.
To find tan 2θ, we use the double-angle formula for tangent:
tan 2θ = (2 tan θ) / (1 - tan² θ)
Substituting tan θ = 3/4:
tan 2θ = (2 * (3/4)) / (1 - (3/4)²)
tan 2θ = (3/2) / (1 - 9/16)
tan 2θ = (3/2) / ((16/16) - (9/16))
tan 2θ = (3/2) / (7/16)
tan 2θ = (3/2) * (16/7)
tan 2θ = (3 * 16) / (2 * 7)
tan 2θ =
tan 2θ = 24/7
Thus, tan 2θ = 24/7.
UNIT SEVEN
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
, 2
Graph: A triangle with vertices labelled A, B, and C. The side opposite angle A has a length of 8. The side opposite
angle B has a length of 6. The angle at vertex C is the right angle, indicated by a small square. The angle at vertex
A is labelled as α, and the angle at vertex B is labelled as θ.
1. Calculate tan 2θ
From the given triangle, we can identify the sides relative to angle θ (angle at B). The side opposite θ is the vertical
side, which has a length of 6. The side adjacent to θ is the horizontal side, which has a length of 8. The hypotenuse
is the remaining side, which we can calculate using the Pythagorean theorem but is not needed for the tangent ratio.
Therefore, tan θ = opposite/adjacent = 6/8 = 3/4.
To find tan 2θ, we use the double-angle formula for tangent:
tan 2θ = (2 tan θ) / (1 - tan² θ)
Substituting tan θ = 3/4:
tan 2θ = (2 * (3/4)) / (1 - (3/4)²)
tan 2θ = (3/2) / (1 - 9/16)
tan 2θ = (3/2) / ((16/16) - (9/16))
tan 2θ = (3/2) / (7/16)
tan 2θ = (3/2) * (16/7)
tan 2θ = (3 * 16) / (2 * 7)
tan 2θ =
tan 2θ = 24/7
Thus, tan 2θ = 24/7.
