sin190cos225tan390 over cos100sin135 =-1 over root 3

Question

I am busy going over some questions from my current maths topic but I'm stuck on this. the question states not to use a calculator, so do I use the reduction formulae if so, how?

Answer 1

This is a proof problem, which means the left side must equal the right side.  The angles are of very large value.  In this case we write the angles as a sum of angles using angles that we can easily evaluate sine and cosine of as our reference angles. Let's rewrite the equation using the sum and difference identities.

 

Here is what we know:

 

cos(225) = cos(180 + 45)

             = cos(180)cos(45) - sin(180)sin(45)

             = -√(2) / 2 

 

sin(135) = sin(180 - 45)

             = sin(180)cos(45) - cos(180)sin(45)

             = -(-√(2) / 2)

             = √(2) / 2 

 

tan(390) = tan(360 + 30)

              = (sin(360 + 30)) / (cos(360 + 30))

              = (sin(360)cos(30) + cos(360)sin(30)) / (cos(360)cos(30) - sin(360)sin(30))

              = (1/2) / (√(3)/2)

              = √(3)/3

 

 

[sin190(-√(2) / 2)(√(3)/3)] / [cos100(√(2) / 2)] = -1/√(3)

 

[sin190(-√(6)/6)] / [cos100(√(2)/2)] = -1/√(3)

 

 

Now we do the same thing to  sin(190)  and  cos(100).  These, however are not that obvious, but if we apply the identities, we can get terms that might cancel each other out later in the process.

 

sin(190) = sin(180 + 10)

             = sin(180)cos(10) + cos(180)sin(10)

             = -sin10

   

 

cos(100) = cos(90 + 10)

             = cos(90)cos(10) - sin(90)sin(10)

             = -sin(10)

 

 

 

Notice that sin(190) = cos(100) = -sin(10).  This is what we want to happen.  Continuing to substitute into equation.

 

 

[(-sin(10))(-√(6)/6)] / [(-sin(10))(√(2)/2)] = -1/√(3)

 

 

 

Now we can simplify.

 

[√(6)sin(10) / 6] * [2 / -√(2)sin(10)] = -1/√(3)

 

(1/3)(-√3) = -1/√(3)

 

-√(3) / 3 = -1/√(3)

 

 

The right side of equation can be rationalized.  The right side becomes -√(3) / 3.

 

Thus, the left side and right side equal each other.

 

 

-

 

 

 

 

Answer 2

For this problem, you'll want to use the Trigonometric Addition Formulas:

 

sin(A+B) = sinAcosB + sinBcosA

cos(A+B) = cosAcosB - sinAsinB

tan(A+B) = (tanA + tanB) / (1 - tanAtanB)

 

First, we'll change each of those ugly-looking numbers from the initial problem into a pair of angles you're familiar with. Notice that there are a couple that don't work out nicely (sin190 and cos100). That's okay. They'll wind up canceling each other out.

 

sin(180+10)cos(180+45)tan(360+30) / cos(90+10)sin(90+45)

 

Now, we'll apply the addition formulas. A warning: This will not be pretty. Write this out on your own piece of paper to make sure you can follow along.

 

((sin180cos10+sin10cos180)(cos180cos45-sin180sin45)(tan360+tan30 / 1-tan360tan30)) / ((cos90cos10-sin90sin10)(sin90cos45+sin45cos90))

 

Finally, let's plug in values you know from the unit circle and simplify:

 

((0×cos10-1×sin10)(-1×^√2/₂-0×^√2/₂)(0+^√3/₃ / 1-0×^√3/₃)) / ((0×cos10-1×sin10)(1×^√2/₂+^√2/₂×0))

((-sin10)(-^√2/₂)(^√3/₃ / 1)) / ((-sin10)(^√2/₂))

(^√2/₂×^√3/₃×sin10) / (-^√2/₂sin10)

 

Crossing out the ^√2/₂ and the sin10 from top and bottom gives you simply -^√3/₃.

 

Glance over at the right-hand side of the equation now. We can change -¹/_√3 by multiplying by ^√3/_√3:

 

-¹/_√3 × ^√3/_√3 = -^√3/₃

 

Thus, the two sides are indeed equal.