Question

Use the trigonometric ratios since, cosine, and tangent to answer the following questions.
Part 1: Find the sine, cosine, and tangent ratios of ∠X.

Part 2: Find the sine, cosine, and tangent ratios of ∠Y.

Part 3: Using your answers from parts 1 and 2, what is significant about the sin∠X and the cos∠Y? Explain your answer.

Part 4: Using your answer from parts 1 and 2, how are the tangents of ∠X and ∠Y related to each other? Explain your answer.

Answer 1

Answer:

Step-by-step explanation:

Part 1.

The sine of an angle is defined as; Opposite side/the Hypotenuse. From the right angle triangle given; sin x =5/13.

The cosine of an angle is defined as; Adjacent side/Hypotenuse. From the right angle triangle given; cos x =12/13.

The tangent of an angle is defined as; Opposite side/Adjacent side. From the right angle triangle given; tan x =5/12.

Part 2.

We use the same definitions of the trigonometric ratios above;

sin y =12/13

cos y =5/13

tan y =12/5

Part 3.

The sin∠X and the cos∠Y are equal, their value is 5/13. The sine of angle is always equal to the cosine of its complement. In this case, ∠X+∠Y =90 hence ∠X is a complement of ∠Y.

Part 4.

The tangents of ∠X and ∠Y are inverses of each other. The tangent of an angle will always be equal to the inverse or reciprocal of the tangent of its complement.

Answer 2

To solve such problems we need to know about Trigonometric functions.

Trigonometric functions

$\rm{Sine\ \theta =\dfrac{Perpendicular}{Hypotenuse}}$

$\rm{Cosine\ \theta =\dfrac{Base}{Hypotenuse}}$

$\rm{tangent\ \theta=\dfrac{Perpendicular}{Base}}$

where perpendicular is the side of the triangle which is opposite to the angle $\bold{ \theta}$, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.

Explanation

Sol. 1: For ∠X,

perpendicular is YZ, P = 5,

base is XZ, B = 12,

hypotenuse is XY, H = 13,

$\rm{Sine (\bold x) =\dfrac{Perpendicular}{Hypotenuse}} = \dfrac{5}{13}}$,

$\rm {Cosine (\bold x) =\dfrac{Base}{Hypotenuse}} = \dfrac{12}{13}}$,

$\rm{Tangent (\bold x) =\dfrac{Perpendicular}{Base}} = \dfrac{5}{12}}$,

Sol. 2: For ∠Y,

perpendicular is XZ, P = 12,

base is YZ, B = 5,

hypotenuse is XY, H = 13,

$\rm{Sine (\bold Y) =\dfrac{Perpendicular}{Hypotenuse}} = \dfrac{12}{13}}$,

$Cosine (\bold Y) =\dfrac{Base}{Hypotenuse}} = \dfrac{5}{13}$,

$Tangent (\bold Y) =\dfrac{Perpendicular}{Base}} = \dfrac{12}{5}$,

Sol. 3:

As we can see above the values of sin∠X and the cos∠Y are equal, which is $\dfrac{5}{13}$. The sine of an angle is always equal to the cosine of its complement in a right-angle triangle. Thus, ∠X+∠Y =90°.

hence, we can say ∠X and ∠Y are complementary angles for each other.

Sol. 4:

As we can see above the values of the tangents of ∠X and ∠Y are inverses of each other, therefore, the values are reciprocal of each other. meaning the tangent of an angle will always be the reciprocal of the tangent of its complementary angle.

Learn more about Trigonometric functions:

https://brainly.com/question/21286835