Question

What is the domain of validity for csctheta =start fraction 1 over sine theta end fraction? (1 point) all real numbers all real numbers except odd multiples of the fraction states pi over 2. all real numbers except even multiples of the fraction states pi over 2. all real numbers except multiples of pi 2. which identity is not used in the proof of the identity 1 + cot2theta = csc2theta? 1 + cot2theta = csc2theta 1 +cos squared theta over sin squared theta= csc2theta sin squared theta over sin squared theta+cos squared theta over sin squared theta= csc2theta sin squared theta plus cos squared theta over sin squared theta= csc2theta one over sin squared theta=csc2theta csc2theta = csc2theta (1 point) cotangent identity pythagorean identity reciprocal identity tangent identity?

Answer 1

Algebra 2b U7 L8 Trigonometric Identities

1.D

2.B

3.D

4.C

5.B

100%

Explanation:

Answer 2

We know the following relationship:


`csc(\theta)=(1)/(sin(\theta))`

The domain of a function are the inputs of the function, that is, a function
`f` is a relation that assigns to each element
`x` in the set A exactly one element in the set B. The set A is the domain (or set of inputs) of the function and the set B contains the range (or set of outputs).Then applying this concept to our function
`csc(\theta)` we can write its domain as follows:

1. Domain of validity for
`csc(\theta)`:


`D: \{\theta \in R/ sin(\theta) \\eq 0 \} \\ In words: All \ \theta \ that \ are \ real \ values \ except \ those \ that \ makes \ sin(\theta)=0`

When:


`sin(\theta)=0`?

when:


`\theta=..., -2\pi,-\pi,0,\pi,\2pi,3pi,...,k\pi`

where k is an integer either positive or negative. That is:


`sin(k\theta)=0 \ for \ k=...,-2,-1,0,1,2,3,...`

To match this with the choices above, the answer is:

"All real numbers except multiples of
`\pi`"


2. which identity is not used in the proof of the identity
`1+cot^(2)(\theta)=csc^(2)(\theta)`:

This identity can proved as follows:


`sin^2{\theta}+cos^(2)(\theta)=1 \ Dividing \ by \ sin^(2)(\theta) \\ \\ \therefore \frac{sin^2{\theta}}{sin^(2)(\theta)}+(cos^(2)(\theta))/(sin^(2)(\theta))=(1)/(sin^(2)(\theta)) \\ \\ \therefore 1+cot^(2)(\theta)=csc^(2)(\theta)`

The identity that is not used is as established in the statement above:

"1 +cos squared theta over sin squared theta= csc2theta"

Written in mathematical language as follows:


`(1+cos^(2)(\theta))/(sin^(2)(\theta))=csc^(2)(\theta)`