Question

find tan j and tan k. write each answer as a fraction in simplest form and as a decimal rounded to four places.
(image of a right triangle jlk with right angle at l, jl=32, lk=24, jk=40)
tan j =
≈
tan k =
≈

Explanation:

Step1: Recall the tangent ratio

In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( J \) and angle \( K \) in right triangle \( \triangle JLK \) (right - angled at \( L \)):

• For \( \tan J \): The side opposite to \( J \) is \( LK = 24 \), and the side adjacent to \( J \) is \( LJ=32 \). So, \( \tan J=\frac{\text{opposite to }J}{\text{adjacent to }J}=\frac{LK}{LJ}=\frac{24}{32} \)
• Simplify \( \frac{24}{32} \): We find the greatest common divisor (GCD) of 24 and 32. The factors of 24 are \( 1, 2, 3, 4, 6, 8, 12, 24 \), and the factors of 32 are \( 1, 2, 4, 8, 16, 32 \). The GCD of 24 and 32 is 8. Divide both the numerator and the denominator by 8: \( \frac{24\div8}{32\div8}=\frac{3}{4} \)
• As a decimal, \( \frac{3}{4}=0.75 \)

Step2: Calculate \( \tan K \)

• For \( \tan K \): The side opposite to \( K \) is \( LJ = 32 \), and the side adjacent to \( K \) is \( LK = 24 \). So, \( \tan K=\frac{\text{opposite to }K}{\text{adjacent to }K}=\frac{LJ}{LK}=\frac{32}{24} \)
• Simplify \( \frac{32}{24} \): The GCD of 32 and 24 is 8. Divide both the numerator and the denominator by 8: \( \frac{32\div8}{24\div8}=\frac{4}{3}\approx1.3333 \) (rounded to four decimal places)

Answer:

• \( \tan J=\frac{3}{4}=0.75 \)
• \( \tan K=\frac{4}{3}\approx1.3333 \)