zalerick:

Hey math side of tumblr, could someone help me with this?

#1.

Look at the formula: sin(2ϴ) = 2sinϴcosϴ

We already know cosϴ, so we just need sinϴ.

Remember SOHCAHTOA, so for cos the ratio is adjacent/hypotenuse. Thus, cosϴ = -3/5 really means that adjacent = 3 and hypotenuse = 5.

The sin ratio is opposite/hypotenuse. We already know hypotenuse = 5, so we just need opposite. Use Pythagorean theorem for right angle triangles.

Side^{2} + OtherSide^{2} = Hypotenuse^{2}

Adjacent^{2} + Opposite^{2} = Hypotenuse^{2}

3^{2} + Opposite^{2} = 5^{2}

9 + Opposite^{2} = 25

Opposite^{2} = 16

Opposite = 4

So, sinϴ = opposite/hypotenuse = 4/5 … but check the sign (+ or -) … Saying ϴ is between 90 and 180 degrees means we are in quadrant 2, where sinϴ is positive. So 4/5 should be kept positive.

Back to the formula sin(2ϴ) = 2sinϴcosϴ = 2(4/5)(-3/5) = -24/25

#2.

This one is similar but we have to do much of the procedure for two different angles, A and B.

Look at the formula: cos(A+B) = cosAcosB - sinAsinB

We already know cosA and sinB, so we just need cosB and sinA.

**Angle A work**

Remember SOHCAHTOA, so for cos the ratio is adjacent/hypotenuse. Thus, cosA = 3/5 really means that adjacent = 3 and hypotenuse = 5.

The sin ratio is opposite/hypotenuse. We already know hypotenuse = 5, so we just need opposite. Use Pythagorean theorem. Same as for #1 above, we get opposite = 4

So, sinA = opposite/hypotenuse = 4/5 … but check the sign (+ or -) … Saying A is between 0 and 90 degrees means we are in quadrant 1, where sin is positive. So 4/5 should be kept positive.

**Angle B work**

SOHCAHTOA again. The sin ratio is opposite/hypotenuse. Thus, sinB = 7/25 really means that opposite = 7 and hypotenuse = 25.

For cos the ratio is adjacent/hypotenuse. We already know hypotenuse = 25, so we just need adjacent. Use Pythagorean theorem.

Adjacent^{2} + Opposite^{2} = Hypotenuse^{2}

Adjacent^{2} + 7^{2} = 25^{2}

Adjacent^{2} = 576

Adjacent = 24

So, cosB = adjacent/hypotenuse = 24/25 … but check the sign (+ or -) … Saying B is between 90 and 180 degrees means we are in quadrant 2, where cos is negative. So make it -24/25.

**All Together Now**

Back to the formula

cos(A+B) = cosAcosB – sinAsinB

= (3/5)(-24/25) – (4/5)(7/25)

= -72/125 – 28/125 = -100/125 = -4/5