math trig question

[nipuna92]

please try to include all the steps

5sin2@ + 12cos2@ = 11

giving all solutions in the interval 0* < @ < 180*

@ means theeta (i dont know how to put it here)
* means degrees

[P3/O/N 2008, (6ii)]

[vanibharutham]

this is a long one

be prepared, it is probably a 13 mark question and is usually breaking down into parts....

(For edexcel C3 By the way)
 

FOR theta i will be USING "x"

5sin2x + 12cos2x = 11

First thing u need to do is to get the left hand side into a common trig ratio

But since we know that sin(A+B) = sinAcosB + sinBcosA,

we can say that

5sin2x + 12cos2x can be written as Rsin(2x+A)

5sin2x + 12cos2x = Rsin(2x+A)
5sin2x + 12cos2x = Rsin2xcosA + RsinAcos2x

Take the sin2x and cos2x from each side and treat them seperately

therefore,

5sin2x = Rsin2xcosA
5 = RcosA
5/cosA = R


and

12cos2x = RsinAcos2x
12 = RsinA

Substituting R from the first part gives us:

12 = 5 sinA/cosA
12/5 = tan A
A = 67.38


R² = 12²+5² (The right angled triangle can be deduced using the earlier formulas in bold)
R = 13

Therefore

5sin2x + 12cos2x = 13sin(2x + 67.38)


Now

13sin(2x+67. = 11

LET 2x + 67.8 be Y

13sinY = 11

Sin Y = 11/13

Y = 57.796, 122.204, 417.796, 482.24
x = -5.352 ,27.02,174.998,207.22

In the range
 x = 27.02,174.998

hope that helped,
if you dont understand something ask away

[nipuna92]

yup i got it

dude this is a cambridge question!!!