MHB Solve Situational Problems Involving Trigonometric Identities

AI Thread Summary
The discussion focuses on solving situational problems involving trigonometric identities, specifically determining the values of sine and cosine for a given angle. The Pythagorean theorem is applied to establish that the hypotenuse is √106, with the opposite side being -5 and the adjacent side being 9. Calculations yield that sin(θ) equals -5/√106 and cos(θ) equals 9/√106. The final expression for sin(θ) + cos(θ) is presented as (9 - 5)/√106. The thread emphasizes the importance of correctly identifying the sides of the triangle to solve the problem effectively.
ukumure
Messages
5
Reaction score
0
Hi! I am so confused about the given and what is being asked, I don't know how to solve it. This topic is solving situational problems involving trigonometric identities. Your help would be a big one for me :) Thank you so much in advance!
1.png
 
Mathematics news on Phys.org
First, we need to establish $\sin\theta$ and $\cos\theta$.
$9^2+(-5)^2=106$ (Pythagorean theorem)
so $\sin\theta$ is $\sqrt{\frac{|-5|}{106}}, \text{that is}, \left(\frac{opp}{hyp}\right)$ and $\cos\theta$ is $\frac{3}{\sqrt{106}}, \text{that is}, \left(\frac{adj}{hyp}\right)$ (recall that $\sin\theta$ is the magnitude of the opposite side of the right-angled triangle containing $\theta$ divided by the hypotenuse)

Hence $\sin\theta+\cos\theta=\frac{3+\sqrt{|-5|}}{\sqrt{106}}$.
 
Greg said:
First, we need to establish $\sin\theta$ and $\cos\theta$.
$9^2+(-5)^2=106$ (Pythagorean theorem)
so $\sin\theta$ is $\sqrt{\frac{|-5|}{106}}, \text{that is}, \left(\frac{opp}{hyp}\right)$ and $\cos\theta$ is $\frac{3}{\sqrt{106}}, \text{that is}, \left(\frac{adj}{hyp}\right)$ (recall that $\sin\theta$ is the magnitude of the opposite side of the right-angled triangle containing $\theta$ divided by the hypotenuse)

Hence $\sin\theta+\cos\theta=\frac{3+\sqrt{|-5|}}{\sqrt{106}}$.
THANK YOU SO MUCH! :)))
 
$\cos{\theta} = \dfrac{x}{r} = \dfrac{9}{\sqrt{106}}$

$\sin{\theta} = \dfrac{y}{r} = \dfrac{-5}{\sqrt{106}}$

$\cos{\theta} + \sin{\theta} = \dfrac{4}{\sqrt{106}}$

ref_tri_IV.jpg
 
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...
Thread 'Imaginary Pythagorus'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Back
Top