If tan 2x = - 24/7, where 90 degrees < x < 180 degrees, then find the value of sin x+ cos x.
I applied various identities and tried manipulating the problem to get sin x + cos x = sin(arctan(-24/7)/2) + cos(arctan(-24/7)/2)
I also played around with the half-angle formula and the double angle formula but they both don't seem to work. Help would be appreciated, thanks.
2θ is in QIV, so θ is in QII, and we have
y = -24
x = 7
r = 25
sin2θ = y/r = -24/25
cos2θ = /r = 7/25
now just apply the half-angle formulas and the identity should fall right out
Thanksa lot
To find the value of sin x + cos x, we can start by finding the value of sin x and cos x separately using the given information.
Given that tan 2x = -24/7, we can use the identity for tan 2x:
tan 2x = (2 tan x) / (1 - tan^2 x)
Plugging in the given value:
-24/7 = (2 tan x) / (1 - tan^2 x)
Let's solve this equation to find the value of tan x. Multiply both sides by (1 - tan^2 x):
-24/7 (1 - tan^2 x) = 2 tan x
Expand the equation:
-24/7 + 24/7 tan^2 x = 2 tan x
Moving terms around:
24/7 tan^2 x - 2 tan x - 24/7 = 0
Now, we can treat this as a quadratic equation in terms of tan x. Let's replace tan x with a variable, such as t:
24/7 t^2 - 2t - 24/7 = 0
Multiply both sides by 7 to clear the fraction:
24 t^2 - 14 t - 24 = 0
Now we can solve this quadratic equation. Factoring might not give us easy whole number solutions, so we will use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 24, b = -14 and c = -24:
t = (-(-14) ± √((-14)^2 - 4 * 24 * -24)) / (2 * 24)
t = (14 ± √(196 + 2304)) / 48
t = (14 ± √2500) / 48
t = (14 ± 50) / 48
We consider both cases:
1. When t = (14 + 50) / 48 = 64 / 48 = 4/3
2. When t = (14 - 50) / 48 = -36 / 48 = -3/4
Using the unit circle, we can see that when 90 degrees < x < 180 degrees, sin x < 0 and cos x < 0. Therefore, we can choose the negative solution t = -3/4.
So, tan x = -3/4. From the unit circle or the trigonometric functions definition, we can determine the values of sin x and cos x.
sin x = -3/5 (opposite/hypotenuse)
cos x = -4/5 (adjacent/hypotenuse)
Finally, we can find sin x + cos x:
sin x + cos x = (-3/5) + (-4/5) = -7/5
Therefore, sin x + cos x = -7/5.