Evaluate 2 tan 240 degrees + 3 cos 12 degrees leaving your answer in surd form
2*Tan240 + 3*Cos12 = 3.46 + 2.93 =
To evaluate 2 tan 240 degrees + 3 cos 12 degrees and express the answer in surd form, we need to use trigonometric identities and reference angles.
First, let's start with the angle 240 degrees. We can find the reference angle by subtracting the nearest multiple of 180: 240 - 180 = 60 degrees. The reference angle for 240 degrees is 60 degrees.
Next, let's consider the angle 12 degrees. In this case, the reference angle is the angle itself because it is less than 90 degrees.
Now, let's evaluate each term separately:
1. Tangent (tan):
Since the reference angle for 240 degrees is 60 degrees, we can use the following identity: tan(240 degrees) = tan(240 degrees - 180 degrees) = tan(60 degrees).
The tangent of 60 degrees is equal to √3. Therefore, 2 tan 240 degrees = 2√3.
2. Cosine (cos):
The cosine of 12 degrees is simply cos(12 degrees).
So, we have 2 tan 240 degrees + 3 cos 12 degrees = 2√3 + 3 cos(12 degrees).
The answer is expressed in surd form for the term 2 tan 240 degrees, and the term 3 cos 12 degrees remains unchanged.
Since it asked for "surd form", they would want exact values
2tan240 is easy
= 2tan60
= 2√3
3 cos 12
= 3cos(30-18)
= 3(cos30cos18 + sin30sin18)
we know cos30 = √3/2 and sin30 = 1/2, so that leaves the 18° angle
There are some interesting ways, find cos18 and sin18
Here is an algebraic way
let θ = 18°
5θ = 90°
2θ + 3θ = 90°
2θ = 90-3θ
sin(2θ) = sin(90-3θ)
sin 2θ = cos 3θ
2sinθcosθ = 4cos^3 θ - 3cosθ <---- basic identities
divide by cosθ and rearrange:
4cos^2 θ - 2sinθ - 3 = 0 , (since cosθ or cos18° ≠ 0, we can do that)
4(1 - sin^2 θ) - 2sinxθ - 3 = 0
4sin^2 θ + 2sinθ - 1 = 0
sin θ = sin18° = (-1 + √5)/4
and cos18° = √( 1 - (√5 - 1)^2 / 16 ) = √(10+2√5)/4
2 tan 240°+ 3 cos 12°
= 2√3 + 3(cos30cos18 + sin30sin18)
= 2√3 + 3( (√3/2) * √(10+2√5)/4 + (1/2) * (-1 + √5)/4 )
there!!! in surd form, whewww!!
Well, well, well, look who's back with some mathematical mischief! Let's break it down step by step, shall we?
First, let's tackle 2 tan 240 degrees. Now, tan of 240 degrees is like finding a needle in a haystack while wearing a blindfold. It's a bit of a challenge. But fear not, my surd-loving friend! From your trigonometry toolbox, you know that tan 240 degrees is equivalent to tan (240 degrees - 180 degrees), right? And what's tan (240 degrees - 180 degrees)? It's the same as tan 60 degrees! And guess what? tan 60 degrees is √3.
So, we can simplify 2 tan 240 degrees to 2√3.
Now, onto the next part of the equation: 3 cos 12 degrees. Ah, cosine! The chill cousin of sine, always there to give us some cool vibes. Now, cos 12 degrees might not be as tricky as tan 240 degrees, but it's still a little sly. However, I'm here to reveal its secret! Hold on tight, because cos 12 degrees is √[(1 + cos 24 degrees)/2].
And now, you can calculate 3 cos 12 degrees by multiplying 3 with √[(1 + cos 24 degrees)/2].
And voila! That's your final answer in surd form: 2√3 + 3√[(1 + cos 24 degrees)/2].
Well done for taking on this mathematical challenge! Remember, humor and a little bit of silliness can help make math more fun!