Question

Angelica says the tangent is the ratio of the length of the leg opposite ZA over the length of the leg

Answer 1

Answer:

1. That the values of the numerator and denominator of tan(A) are the original side lengths of opposite and adjacent legs to ∠A of the triangle

2. That it is only with the actual side lengths that the trigonometric ratios can be derived even when that is that the trigonometric ratios of sine and cosine cannot be derived from the other known trigonometric ratio such as the trigonometric ratio of the tangent

3. cos(A) is 4/5

sin(A) is 3/5

Step-by-step explanation:

Given that tan(A) = 3/4

Angela says the tangent = (The length of leg opposite ∠A)/(The length of leg adjacent to ∠A)

Therefore;

The length of leg opposite ∠A = 3

The length of leg adjacent to ∠A = 4

From where AB = √(3² + 4²) = 5

Graciella says that they cannot find the length and therefore cannot find the trigonometric ratios

1. The mistake Angela made was taking the value of the tangent which is the ratio of two side length to be the actual side lengths in the division

An example is given by assuming we have the following side length values;

The length of leg opposite ∠A = 15

The length of leg adjacent to ∠A = 20

The tangent = (The length of leg opposite ∠A)/(The length of leg adjacent to ∠A)

∴ The tangent = 15/20 = 3/4 as before, but the actual side lengths are 15 and 20 making a conclusion that the actual side length of the triangle are given by the values of the numerator and denominator of the trigonometric ratio to be in error

2. The mistake Graciella made in her reasoning was that he actual values of the side lengths rather than their specific ratio with each other are required to find the trigonometric ratios of sine and cosine

3. Given that the relationship between tan(A) and cos(A) is given as follows;

sec²(A) - tan²(A) = 1

Where;

$sec(A) = \dfrac{1}{cos(A)}$

Therefore;

sec²(A) - (3/4)² = 1

sec²(A) = 1 + (3/4)² = 25/16

cos²(A) = 1/(25/16) = 16/25

cos(A) = 4/5

Also we have;

sin²(A) + cos²(A) = 1

Which gives;

sin²(A) + 16/25 = 1

sin²(A)  = 1 - 16/25 = 9/25

sin(A) = 3/5.