What is sin(α + β)

Question

If 180° < α < 270°, cos α= -8/17, 270° < β < 360°, and sin β= -4/5, what is sin(α + β)?

Answer 1

Draw a figure and use the theorem of Pythagoras to find the missing sides.

cos α=-8/17 means sin α = -15/17

sin β =- 4/5 means cos β = 3/5

Now use the formula for the sin of the sum of 2 angles to get your result...BUT be careful with signs!

Answer 2

Given:

cos A = -8/17, in the 3rd quadrant

sin B = -4/5, in the 4th quadrant

Find: sin (A+B)

We want to use this sum of 2 angles identity:

sin(A+B) = sin A cos B + cos A sin B

Use Pythagorean Theorem to find sin A and cos B:

sin A = +- sqrt( (17^2 - 8^2) / 17^2 ) = -15/17 (3rd quadrant)

Similarly,

cos B = +3/5 (4th quadrant)

Now we just plug into the identity:

sin(A+B) = -15/17 * 3/5 + (-8/17) * (-4/5) = -13/85 (exact)

Answer 3

The first angle is 241.93 degrees & the second angles 306.9 degrees

then sin(241.93 + 306.9) = sin(241.93)cos(306.9) + cos(241.93)sin(306.9)

= (-.8824)(.6) + (-.4705)(-.8) = -.153