1. ## Trigonometric Expression

Simplify the trigonometric expression.

csc 40°(sin 40° + sin 80° + sin 120°).

The answer is 4 cos^2 (10°).

My Work:

I have no idea how to reach the answer without using my calculator.

csc 40° = 1/sin 40°

1/sin 40° = 1.5557238268

sin 40° + sin 80° + sin 120° = 2.4936207664

Multiply:

(1.5557238268)(2.4936207664) =

3.8793852412 which equals

4 cos^2 (10°).

How can I reach the same answer without use of my calculator?

Thank you.

2. ## Re: Trigonometric Expression

You have sin(40)= sin(4(10)), sin(80)= sin(8(10)), and sin(120)= sin(12(10)).

So I would recommend some trig identities:
sin(x+ y)= sin(x)cos(y)+ cos(x)sin(y)
cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

Setting x= y= t, sin(2t)= 2 sin(t)cos(t) and cos(2t)= cos^2(t)- sin^2(t).

sin(3t)= sin(2t+ t)= sin(t)cos(2t)+ cos(t)sin(2t)= sin(t)(cos^2(t)- sin^2(t))+ cos(t)(2sin(t)cos(t))= sin(t)cos^2(t)- sin^3(t)+ 2sin(t)cos^2(t)= 3sin(t)cos^2(t)- sin^3(t).
With t= 40, sin(80)= 2 sin(40)cos(40) and sin(120)= 3 sin(40)cos^2(40)- sin^3(40).

With t= 10, sin(20)= 2sin(10)cos(10) and cos(20)= cos^2(10)- sin^2(10).
sin(40)= 2sin(20)cos(20)= 2sin(10)cos(10)(cos^2(10)- sin^2(10))= 2sin(10)cos^3(10)- 2sin^3(10)cos(10) and
cos(40)= cos^2(20)- sin^2(20)= (cos^2(10)- sin^2(10))^2- (2sin(10)cos(10)^2= cos^4(10)- 2cos^2(10)sin^2(10)+ sin^4(10)- 4sin^2(10)cos^2(10)=
cos^4(10)- 6cos^2(10)sin^2(10)+ sin^4(10)

Keep going!

3. ## Re: Trigonometric Expression

Originally Posted by HallsofIvy
You have sin(40)= sin(4(10)), sin(80)= sin(8(10)), and sin(120)= sin(12(10)).

So I would recommend some trig identities:
sin(x+ y)= sin(x)cos(y)+ cos(x)sin(y)
cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

Setting x= y= t, sin(2t)= 2 sin(t)cos(t) and cos(2t)= cos^2(t)- sin^2(t).

sin(3t)= sin(2t+ t)= sin(t)cos(2t)+ cos(t)sin(2t)= sin(t)(cos^2(t)- sin^2(t))+ cos(t)(2sin(t)cos(t))= sin(t)cos^2(t)- sin^3(t)+ 2sin(t)cos^2(t)= 3sin(t)cos^2(t)- sin^3(t).
With t= 40, sin(80)= 2 sin(40)cos(40) and sin(120)= 3 sin(40)cos^2(40)- sin^3(40).

With t= 10, sin(20)= 2sin(10)cos(10) and cos(20)= cos^2(10)- sin^2(10).
sin(40)= 2sin(20)cos(20)= 2sin(10)cos(10)(cos^2(10)- sin^2(10))= 2sin(10)cos^3(10)- 2sin^3(10)cos(10) and
cos(40)= cos^2(20)- sin^2(20)= (cos^2(10)- sin^2(10))^2- (2sin(10)cos(10)^2= cos^4(10)- 2cos^2(10)sin^2(10)+ sin^4(10)- 4sin^2(10)cos^2(10)=
cos^4(10)- 6cos^2(10)sin^2(10)+ sin^4(10)

Keep going!
This problem takes too long to type. It is super tedious.

4. ## Re: Trigonometric Expression

$\sin 80{}^{\circ}+\sin 40{}^{\circ} + \sin 120{}^{\circ} =\sin 80{}^{\circ}+2 \sin 80{}^{\circ} \cos 40{}^{\circ}=$

$\sin 80{}^{\circ}(1+2 \cos 40{}^{\circ})$

5. ## Re: Trigonometric Expression

Originally Posted by Idea
$\sin 80{}^{\circ}+\sin 40{}^{\circ} + \sin 120{}^{\circ} =\sin 80{}^{\circ}+2 \sin 80{}^{\circ} \cos 40{}^{\circ}=$

$\sin 80{}^{\circ}(1+2 \cos 40{}^{\circ})$
Can you please complete this problem for me? It will help me solve similar questions in my textbook.

6. ## Re: Trigonometric Expression

$\frac{\sin 80 +(\sin 40 + \sin 120)}{\sin 40}=$

$\frac{\sin 80 (1+ 2 \cos 40)}{\sin 40}=$

$4 \cos 40 \left(\frac{1}{2} +\text{ }\cos 40\right)=$

$4 \cos 40 ( \cos 60 + \cos 40)=$

$4 \cos 40\text{ }2 \cos 50 \cos 10=$

$8 \sin 50 \cos 50 \cos 10=$

$4 \sin 100 \cos 10=$

$4 \cos ^2 10$

7. ## Re: Trigonometric Expression

Originally Posted by Idea
$\frac{\sin 80 +(\sin 40 + \sin 120)}{\sin 40}=$

$\frac{\sin 80 (1+ 2 \cos 40)}{\sin 40}=$

$4 \cos 40 \left(\frac{1}{2} +\text{ }\cos 40\right)=$

$4 \cos 40 ( \cos 60 + \cos 40)=$

$4 \cos 40\text{ }2 \cos 50 \cos 10=$

$8 \sin 50 \cos 50 \cos 10=$

$4 \sin 100 \cos 10=$

$4 \cos ^2 10$
Thank you so much.