Find the exact value of cos[cot^-1 (-√3) + sin^-1 (-1/2)].
I'm having trouble with inverses. Please help by showing work.
cos[cot^-1 (-√3) + sin^-1 (-1/2)]
let's take it in parts
cot^-1 (-√3)
is the angle so that cotØ = -√3
or tanØ = -1/√3
I know tan30° = +1/√3
so Ø = 180-30 = 150° or -30°
(usually we take the smallest positive angle)
sin^-1 (-1/2)
= 180+45 = 225°
cos[cot^-1 (-√3) + sin^-1 (-1/2)]
= cos(150 + 225)
= cos 375°
= cos(360 + 15)
= cos15
= cos(45-30)
= cos45cos30 + sin45sin30
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4
just for ease of readability, let's say
x = cot^-1 (-√3)
y = sin^-1 (-1/2)
The principal values of these inverse trig functions will be in QIV, so draw the triangles there. Then you can see that
sinx = -1/2
cosx = √3/2
siny = -1/2
cosy = √3/2
cos(x+y) = cosx cosy - sinx siny
= √3/2 * √3/2 - 1/2 * 1/2
= 3/4 - 1/4
= 1/2
Or, you could just recognize that
x = y = -π/6
so cos(x+y) = cos(-π/3) = 1/2
go with Steve's answer,
I forgot that we could pin-point the quadrant (although mine had the more-fun calculations)
Thanks so much! I'm starting to get it. :)
To find the exact value of the expression cos[cot^⁻1(-√3) + sin^⁻1(-1/2)], we'll follow these steps:
Step 1: Identify the angles
Let's start by finding the values of cot^⁻1(-√3) and sin^⁻1(-1/2).
cot^⁻1(-√3):
cot^⁻1 means the angle whose cotangent is -√3. We can rewrite it as arccot(-√3).
Two common trigonometric identity formulas will be helpful here:
cot(x) = 1/tan(x)
cot^⁻1(x) = arccot(x) = atan(1/x)
Applying atan(1/x) to arccot(-√3), we have:
arccot(-√3) = atan(1/(-√3))
Now, let's rationalize the denominator:
arccot(-√3) = atan(-√3/3)
We can determine the reference angle of atan(-√3/3) by using the Pythagorean identity.
Let's denote the reference angle as A:
tan(A) = √3/3
Using a 30-60-90 triangle or the unit circle, we find that the reference angle A is π/6.
So, arccot(-√3) = atan(-√3/3) = -π/6
sin^⁻1(-1/2):
sin^⁻1 means the angle whose sine is -1/2. We can rewrite it as arcsin(-1/2) = -π/6.
Step 2: Substitute values into the expression
We now have cos(-π/6 + (-π/6)).
Step 3: Simplify the expression
cos(-π/6 + (-π/6)) = cos(-2π/6) = cos(-π/3)
Step 4: Use the properties of cosine function
cos(-π/3) = cos(π - π/3) = cos(2π/3)
Step 5: Find the exact value
Using the unit circle or reference angles, we know that cos(2π/3) = -1/2.
Therefore, the exact value of cos[cot^⁻1(-√3) + sin^⁻1(-1/2)] is -1/2.