Question

Refer to the diagram at the right.

a. Express the sine, cosine, tangent of a in terms of x,y, and z
b. Express the sine, cosine, and tangent of ẞ in terms of
x, y, and z.
c. Tell whether each of the following statements is true
or false.
1) sin a=cosb 2) tan a =cot b 3) sec a =csc B

Answer 1

Answer:

See below

Step-by-step explanation:

To express the trigonometric functions of angles $\alpha$ and $\beta$ in terms of the given sides $x$, $y$, and $z$, we can use the definitions of sine, cosine, and tangent:

a) For angle $\alpha$:

$\sin \alpha =\dfrac{\textsf{opposite}}{\textsf{hypotenuse}} \\\\ = \dfrac{x}{z}$

$\cos \alpha = \dfrac{\textsf{adjacent}}{\textsf{hypotenuse}} \\\\\ = \dfrac{y}{z}$

$\tan \alpha =\dfrac{\textsf{opposite}}{\textsf{adjacent}} \\\\ = \dfrac{x}{y}$

b) For angle $\beta$:

$\sin \alpha =\dfrac{\textsf{opposite}}{\textsf{hypotenuse}} \\\\ = \dfrac{y}{z}$

$\cos \alpha = \dfrac{\textsf{adjacent}}{\textsf{hypotenuse}} \\\\\ = \dfrac{x}{z}$

$\tan \alpha =\dfrac{\textsf{opposite}}{\textsf{adjacent}} \\\\ = \dfrac{y}{x}$

c)

Now, let's determine whether each of the given statements is true or false:

1) $\sin \alpha = \cos \beta$:

$\sin \alpha = \dfrac{x}{z}$

$\cos \beta = \dfrac{x}{z}$

Since both expressions are equal, this statement is true.

2) $\tan \alpha = \cot \beta$:

$\tan \alpha = \dfrac{x}{y}$

$\cot \beta = \dfrac{1}{\tan \beta} = \dfrac{1}{\dfrac{y}{x}} = \dfrac{x}{y}$

Since both expressions are equal, this statement is true.

3) $\sec \alpha = \csc \beta$:

$\sec \alpha = \dfrac{1}{\cos \alpha} = \dfrac{1}{\dfrac{y}{z}} = \dfrac{z}{y}$

$\csc \beta = \dfrac{1}{\sin \beta} = \dfrac{1}{\dfrac{y}{z}} = \dfrac{z}{y}$

Since both expressions are equal, this statement is true.

Therefore, all three statements are true.