MHB Angle Sum/Difference Identities: Billy's Pre-calc Math Problem

AI Thread Summary
The discussion revolves around solving a pre-calculus problem involving angle sum and difference identities for tangent. Given cos(a) = 15/17 and csc(B) = 41/9, the tangent values for angles a and B are calculated as tan(a) = 8/15 and tan(B) = 9/40, respectively. Using the tangent identities, the results for tan(a + B) and tan(a - B) are found to be 455/528 and 185/672. The solution emphasizes the application of Pythagorean identities to derive the necessary tangent values. This discussion serves as a resource for those seeking help with similar trigonometry problems.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Pre-calc math problem?

a and B are quadrent I angles with cos(a) = 15/17 and csc(B) = 41/9.

find tan (a + B) and tan (a-B)

Here is a link to the question:

Pre-calc math problem? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Re: Billy's question from Yahoo! Answers involving the angle sum/difference identities for tangent

Hello Billy,

We are given:

$$\cos(\alpha)=\frac{15}{17}$$

and using the Pythagorean identity $\tan^2(\alpha)=\sec^2(\alpha)-1$ we find (given $\alpha$ is in the first quadrant, and so all trig. functions are positive there:

$$\tan(\alpha)=\sqrt{\left(\frac{17}{15} \right)^2-1}=\frac{8}{15}$$

We are also given:

$$\csc(\beta)=\frac{41}{9}$$

and using the Pythagorean identity $\cot^2(\beta)=\csc^2(\beta)-1$ we find:

$$\tan(\beta)=\frac{1}{\cot(\beta)}=\frac{1}{ \sqrt{\left(\frac{41}{9} \right)^2-1}}=\frac{9}{40}$$

Now, using the angle sum/difference identity for tangent $$\tan(\alpha\pm\beta)=\frac{\tan(\alpha)\pm\tan( \beta)}{1\mp\tan(\alpha)\tan( \beta)}$$, we find:

$$\tan(\alpha+\beta)=\frac{\frac{8}{15}+\frac{9}{40}}{1-\frac{8}{15}\cdot\frac{9}{40}}=\frac{455}{528}$$

$$\tan(\alpha-\beta)=\frac{\frac{8}{15}-\frac{9}{40}}{1+\frac{8}{15}\cdot\frac{9}{40}}= \frac{185}{672}$$

To Billy and any other visitors reading this topic, I invite you to register and post other trigonometry questions in our http://www.mathhelpboards.com/f12/ forum.
 
Last edited:
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top