Angelica says the tangent is the ratio of the length of the leg opposite ZA over the length of the leg adjacent to ZA. This means BC must have a length of 3 units and AC must have a length of 4 units. They
can find the length of AB using the Pythagorean Theorem and then use it to find the sine and cosine ratios.
Graciella says that they cannot find the length of any side of the triangle and, therefore, cannot find the
ratios for sine and cosine.
Both students have errors in their reasoning.
1. Identify the mistake Angelica made in her reasoning. Provide an example that shows why she is
incorrect.
2. Identify the mistake that Graciella made in her reasoning. Explain why she is incorrect.
3. Find the ratios for the sine and cosine of ZA. Explain your process.

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;

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.

answer: the answer is 40.82

step-by-step explanation: 3.14 x 13(or half of 26) to the power of 2

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