Trig: Basic Identities Problem

In summary, the question asks to find the exact value of tan(2B) given that sin(B) = -5/13. Using the Pythagorean identity and plugging in values for sin(B) and cos(B), we can solve for tan(2B) to get \frac{120}{119} and \frac{-120}{119} as the two possible answers.
  • #1
Nik
4
0
Hi, today I was writing my Identities final test and got stuck on one question which I couldn't solve.

The question was: If [tex]sinB=-\frac{5}{13}[/tex] find the exact value of [tex]tan2B[/tex].

I've got this far, correct me if I went wrong anywhere: [tex]tan2B=\frac{sin2B}{cos2B}=\frac{2sinBcosB}{2cos^2B-1}[/tex]

Then, when I tryied to solve further I came to a dead end, even though I tried many different ways.

Thanx in advance!
 
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  • #2
There should probably be 2 answers.

First of all, with that triangle, the hypotinuse is 13, x is +12 and -12, y is -5, and B is either in the third or fourth quadrant.

Fill in the equation using x as 12, then fill in the equation using -12.
 
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  • #3
Tnx for your help, the answers I've got are [tex]\frac{120}{119}[/tex] and [tex]\frac{-120}{119}[/tex]

Once I get my test back I'll post if the answers match.
 
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  • #4
If sin(B) = -5/13, then sin^2(B) = 25/169. Using the Pythagorean identity, we have that cos^2(B) = 1 - sin^2(B) = 1 - 25/169 = 144/169, which gives cos(B) = +/- 12/13. Plug those values into the formula for tan(2B) and voila...
 

1) What are the basic identities in trigonometry?

The basic identities in trigonometry are the Pythagorean identities, reciprocal identities, quotient identities, and cofunction identities. These identities involve the six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

2) How do you solve trigonometry problems using basic identities?

To solve trigonometry problems using basic identities, you must first identify which identities are relevant to the given problem. Then, use algebraic manipulations and substitution to simplify the expressions and solve for the unknown variables.

3) What is the Pythagorean identity in trigonometry?

The Pythagorean identity in trigonometry is sin^2(x) + cos^2(x) = 1. This identity is derived from the Pythagorean theorem and is used to relate the sine and cosine functions.

4) How do you prove a trigonometric identity?

To prove a trigonometric identity, you must use algebraic manipulations to show that the left side of the equation is equal to the right side of the equation. This can involve using basic identities, trigonometric properties, and substitution of known values.

5) What are some common mistakes when using basic identities in trigonometry?

Some common mistakes when using basic identities in trigonometry include forgetting to use the correct sign for the trigonometric function, not simplifying the expression enough, and using incorrect identities for the given problem. It is important to double check your work and make sure you are using the appropriate identities for the given problem.

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