tan(θ + ϕ); cos(θ) = − 1/3
θ in Quadrant III, sin(ϕ) = 1/4 ϕ in Quadrant II
I am really struggling with this idk why I keep getting the wrong answers
what are all those sin() stuff still hanging around? The final expression should just be a bunch of fractions.
tan(θ + ϕ); cos(θ) = − 1/3
θ in Quadrant III, sin(ϕ) = 1/4 ϕ in Quadrant II
You really need to review the signs of the trig functions in the various quadrants.
In QIII,
sinθ = -√8/3
cosθ = -1/3
In QII,
sinϕ = 1/4
cosϕ = -√15/4
tan(θ+ϕ) = sin(θ+ϕ)/cos(θ+ϕ)
= (sinθcosϕ+cosθsinϕ)/(cosθcosϕ-sinθsinϕ)
= ((-√8/3)(-√15/4)+(-1/3)(1/4))/((-1/3)(-√15/4)-(-√8/3)(1/4))
= (32√2-9√15)/7
Or, using tan(θ+ϕ) = (tanθ+tanϕ)/(1-tanθ*tanϕ)
tanθ = sinθ/cosθ = √8
tanϕ = sinϕ/cosϕ = -1/√15
tan(θ+ϕ) = (√8 - 1/√15)/(1+√8/√15) = (32√2-9√15)/7
i was given this equation Tan(a+b)=sin(a+b)/cos(a+b)
So I did (square root 8/3)(square root 15/4)+sin(1/4)cos(-1/3) divide that by (-1/3)(square root 15 over 4) - sin( square root 8 over 3)sin(1/4)
Don't worry, I'll help you solve this. Let's break it down step by step.
Given:
cos(θ) = -1/3
θ is in Quadrant III
sin(ϕ) = 1/4
ϕ is in Quadrant II
First, let's find the value of tan(θ + ϕ). To do that, we'll need the values of both θ and ϕ.
Since cos(θ) is given as -1/3, we know that the adjacent side length to the angle θ is -1 and the hypotenuse is 3. Using the Pythagorean theorem, we can find the opposite side length. Let's call it y.
Using the formula:
cos(θ) = adjacent/hypotenuse = -1/3
We can rearrange this to find y:
y = sqrt(hypotenuse^2 - adjacent^2) = sqrt(3^2 - (-1)^2) = sqrt(9 - 1) = sqrt(8) = 2√2
Now, let's consider the Quadrant III. In this quadrant, both x and y values are negative. Since we have the opposite side length as positive (2√2), we know that θ is the angle between the negative x-axis and the hypotenuse in Quadrant III.
Next, let's find the value of sin(ϕ). We are given that sin(ϕ) = 1/4. In Quadrant II, the opposite side length is positive. Using the formula sin(ϕ) = opposite/hypotenuse, let's find the adjacent side length. Let's call it x.
Given:
sin(ϕ) = opposite/hypotenuse = 1/4
opposite side = 1, hypotenuse = 4
Using the Pythagorean theorem, we can find the adjacent side length:
x = sqrt(hypotenuse^2 - opposite^2) = sqrt(4^2 - 1^2) = sqrt(16 - 1) = sqrt(15)
In Quadrant II, the x-values are negative, so we have x = -sqrt(15).
Now that we have the values of θ and ϕ, we can calculate tan(θ + ϕ). The formula for tan(θ + ϕ) is:
tan(θ + ϕ) = (tan(θ) + tan(ϕ)) / (1 - tan(θ) * tan(ϕ))
Substituting the values we found, we have:
tan(θ + ϕ) = (tan(θ) + tan(ϕ)) / (1 - tan(θ) * tan(ϕ))
= (2√2 + (-sqrt(15))) / (1 - (2√2)(-sqrt(15)))
Simplifying the expression, we get:
tan(θ + ϕ) = (√2 * (2 - √15)) / (1 + 2√30)
This is the value of tan(θ + ϕ) based on the given conditions.