Help needed with two problems (one a trig question, the other a world prob)

[DLxX]

1. Solve for theta in the interval 0<theta<2pi. Give answers in radians either correct to 3 decimal places or as exact values.

13cos^2 - 5cos = 12sin^2 - 6

3) Scientists use visual magnitude to measure the brighness of a star. The smaller the number, the brighter the star. An increase in magnitude of 1 unit represents a decrease of 60% in the brighness of a star.

a) The star Vega has a magnitude of 0. If a second star has a visual magnitude of -2.4, how many times brighter than Vega is this star?

b) If a star is 15% as bright as Vega is , what is its visual magnitude?

c)How is this scale different from the Richter scale?




For this next question all I need is a basic run through of what I have to do.

4) a and B are both angles in the second quadrant. If sin a = 1/3 and sin B = 2/3, find the exact value of cos (a +b).

Thanks

 

[HallsofIvy]

1. Solve for theta in the interval 0<theta<2pi. Give answers in radians either correct to 3 decimal places or as exact values.

13cos^2 - 5cos = 12sin^2 - 6

The first thing you should really try doing is copying the problem correctly! That is not the problem you were given because it doesn't make sense! I suspect that the problem is 13 cos²θ- 5 cosθ= 12 sin²θ- 6.
Replace sin²&theta with 1- cos²θ and you have a quadratic equation for cosθ solve for cosθ and then then use a calculator.

3) Scientists use visual magnitude to measure the brighness of a star. The smaller the number, the brighter the star. An increase in magnitude of 1 unit represents a decrease of 60% in the brighness of a star.

a) The star Vega has a magnitude of 0. If a second star has a visual magnitude of -2.4, how many times brighter than Vega is this star?

\
A "decrease" of 60% is the same as 40% or 0.4. -2.4 is a decrease in magnitude of 2.4 so an increase in brightness of 0.4 2.4 times: careful: that is not 2.4*0.4 but (0.4)^2.4.

b) If a star is 15% as bright as Vega is , what is its visual magnitude?

"15% as bright" is a decrease of 100-15= 85= (85/40)(40)= (2.125)(40) %

c)How is this scale different from the Richter scale?

Well, how is the Richter scale defined?

For this next question all I need is a basic run through of what I have to do.

4) a and B are both angles in the second quadrant. If sin a = 1/3 and sin B = 2/3, find the exact value of cos (a +b).

If sin a= 1/3 what is cos a? If sin B= 2/3 what is cos B? Do you know a formula for cos(a+b) in terms of cos(a), sin(a), cos(b), sin(b)?

Thanks