Question

In the following right triangle, which one of the following show the value of tan(theta) and sec(theta)?

Triangle 90degree
AB = 4
AC = 6
Angle C = theta
Angle B = 90 degree
A) (2√ 5)/5 and 3/ 2
B) (2√5)/5 and (3√5)/5
C) (√5)/2 and (3√5)/5
D) (√5)/2 and 3/2

Answer 1

Using the Pythagorean theorem, we found that the hypotenuse of the right triangle is 2sqrt(13). The values of tan(theta) and sec(theta) are then calculated to be 3/2 and sqrt(13)/2 respectively, which do not match any of the provided options A to D.

Step-by-step explanation:

To find the value of tan(theta) and sec(theta) in a right triangle where AB = 4, AC = 6, and angle C = theta, we first need to determine the length of the hypotenuse BC using the Pythagorean theorem.

Applying the theorem: AB^2 + AC^2 = BC^2, we get 4^2 + 6^2 = BC^2, which simplifies to 16 + 36 = BC^2, hence BC^2 = 52, and BC = sqrt(52) which simplifies to 2sqrt(13).

Tan(theta) is the ratio of the opposite side to the adjacent side, so tan(theta) = AC/AB which equals 6/4 or 3/2. To find sec(theta), which is the reciprocal of the cosine function, we first find the cosine of theta (cos(theta)) as the adjacent side over the hypotenuse, which is AB/BC, so cos(theta) = 4/(2sqrt(13)). The reciprocal of this, sec(theta), is (2sqrt(13))/4, simplifying to sqrt(13)/2.

Therefore, the values of tan(theta) and sec(theta) are 3/2 and sqrt(13)/2 respectively. None of the provided options A to D are correct based on the calculations.