Question

For the triangle ABC, determine if the following statement is True or False.

Answer 1

Answer:

sin A = cos B is TRUE

csc A = sec A is FALSE

tan A = cot B is TRUE

Step-by-step explanation:

Let's review the basics of trigonometry and what each ratio means. We will base our discussion with respect to the triangle shown in the picture attached.

For any right triangle, the side opposite of the right angle is called the "hypotenuse". The other two sides are legs of the triangle. Those are also called "opposite" side and "adjacent". Which one is which depends on the angle (besides the right angle) we are looking at.

With respect to the angle θ, the "opposite" side would be "b" and the adjacent side would be "a" and of course, c is the hypotenuse. If we were to look from the other angle, then "opposite" would have been "a" and adjacent would have been "b".

Trigonometric Ratios:

6 trigonometric ratios (sin, cos, tan, csc, sec, & cot) are ratios of one side to another. Below are the relationships give:

• Sin θ is the ratio of opposite to hypotenuse
• Cos θ is the ratio of adjacent to hypotenuse
• Tan θ is the ratio of opposite to adjacent
• Csc θ is the ratio of hypotenuse to opposite
• Sec θ is the ratio of hypotenuse to adjacent
• Cot θ is the ratio of adjacent to opposite

Let's solve our problem now:

For our image given in this problem, we will figure out the ratios and see if the 3 equations hold true or not.

Sin A = Cos B

Sin A = $\frac{a}{c}$

Cos B = $\frac{a}{c}$

These two are equal. So this statement is TRUE.

Csc A = Sec A

Csc A = $\frac{c}{a}$

Sec B = $\frac{c}{b}$

These two are NOT equal. So this statement is FALSE.

Tan A = Cot B

Tan A = $\frac{a}{b}$

Cot B = $\frac{a}{b}$

These two are equal. So this statement is TRUE.