trig problems

Question

1. Prove that: [(sin2A - cos2A+1)/ (sin2A + cos2A+1)=tanA 2. Solve for A if: 2 (cos^2) A = -cos A Where 0degrees is less than and equal x less than and equal 360degrees. 3. Derive a formula for cos2A Use the derived formula to determine the value of cos120degress without the use of a calculator. 4. If sin A=0.6 and sin B=0.4 ; and both A and B are acute angles, determine the value of cos(A+B) without a calculator. 5. Prove that: Sin(180+A) = -sinA

Answer 1

1)

(sin2A - cos 2A + 1) / (sin2A + cos 2A + 1) = tanA

Left hand side identity (sin2A - cos 2A + 1) / (sin2A + cos 2A + 1)

Using multiple trignometric identity.

- cos 2A + 1 = 2sin²A; cos 2A + 1 = 2cos²A and  sin2A = 2sinAcosA .

= (2sinAcosA + 2sin²A) / (2sinAcosA + 2cos²A)

=2 sin A (cosA + sinA) / 2cosA(sinA + cosA)

= sin A  / cosA

= tanA

= Right hand side identity.

Answer 2

2)

cos² A = -cos A

cos² A + cos A = 0

cos A (cos A + 1) = 0

cos A = 0 , cos A + 1 = 0.

cos A = 0

cos A = cos (π/2)

The general solution is  A = 2nπ ± (π/2) = π/2 ( 4n ± 1)

At n = 0 => A = π/2

At n = 1 => A = 3π/2

cos A = -1

cos A = cos (π)

The general solution is  A = 2nπ ± (π) = π ( 2n ± 1)

At n = 0 => A = π

Therefore, the solutions are  π/2, π , and 3π/2.

Answer 3

3)

cos2A  = cos ( A + A )

           = cos A cos A - sin A sin A              (Using trigonometric sum formula:

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

          = cos² A - sin² A 

cos120° = cos 2( 60° )

            = cos² 60° - sin² 60° 

             = (1/2)²  - (√3/2)²

               = 1/4  -  3/4

              = -2/4

              = -1/2 

Answer 4

4)

sin A = 0.6

cos A = √( 1 - sin² A)

          = √( 1 - 0.6²)

cos A = 0.8

sin B = 0.4

cos B = √( 1 - sin² A)

          = √( 1 - 0.4²)

cos B = 0.92

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

                = 0.8* 0.92 - 0.6 * 0.4

               = 0.5

Answer 5

5)

sin (180 + A)

Use sum formula : sin (A+B) = sinA cos B + cos A sinB

sin (180 + A) = sin180 cosA + cos 180 sin A

                    = 0 - sin A

                   = -sin A