How Can You Solve These Complex Trigonometry Problems?

[harimakenji]

Homework Statement

1. 3 sin^2 x + 2 sin^2 y = 1 and 3 sin 2x - 2 sin 2y = 0; find sin (x + 2y)
2. sin^3 10 + cos^3 40 - sin^3 70 = -3/8
3. sin P + sin Q + sin R = 2 cos 1/2Q cos 1/2R; find angle P
4. cos A = 2 cos B; find (tan (1/2A + 1/2B))/(tan (1/2A - 1/2B))
5. 1/cos 0 cos 1 + 1/cos 1 cos 2 + ... + 1/cos 88 cos 89 = cos 1/sin^2 1
6. tan 70 - tan 20 (without using calculator)
7. tan B = (tan A + tan C)/(1 + tan A.tan C); show that sin 2B = (sin 2A + sin 2C)/(1 + sin 2A.sin 2C)

Homework Equations

For number 1, 2, 3, 4 and 7, may only use double angle formula, sum and difference formula and half-angle formula (trigonometry identities are also permitted).
For number 5, 6 may use all formulas in trigonometry.

The Attempt at a Solution

I've tried several times and got stuck..
1. i got the simplest form for sin (x + 2y) = 3 sin x. I am not really sure that this is the final answer. Usually we got number for the final answer, not an another form of trigonometry.

2. to answer this, i have to use other formulas which are not permitted by the question. i got -3/8 for the answer. I guess this is true, but i don't know if there's any other way to do it by using only the formulas permitted by the question.

3. got stuck.. really got no idea. there's no info about the relationship between P, Q and R..

4. I've tried to solve this. Unfortunately, i got tan (1/2A + 1/2B).tan (1/2A - 1/2B) instead (which is -3). It does not answer the question though..

5. this one, I've tried and i have finally found the cos 1 (which is what the LHS needed), but i got no idea for the denominator.. there's nothing i could do to get sin^2 1.

6. by using calculator, the result was terrible. i am not sure the question is right.. i got no number for the final answer, it is still in the form of another trigonometry..

7. i managed to find the numerator (which is sin 2A + sin 2C), but the denominator was far different from what the question asked (i got cos^2 A.cos^2 C)

 

[nate808]

so i think it is not the answer.




Hello, thank you for sharing your attempts at solving these problems. I can understand your frustration with some of the questions, as they do require a good understanding of trigonometric identities and manipulation skills. I will try my best to guide you through each problem and hopefully, you will be able to solve them on your own.

1. For this problem, we can use the sum and difference formula for sine and cosine to simplify the equations. We have:
3(sin^2 x + sin^2 y) = 1
3(sin^2 x + (1 - cos^2 y)) = 1 (using the identity sin^2 y + cos^2 y = 1)
3sin^2 x + 3 - 3cos^2 y = 1
3sin^2 x - 3cos^2 y = -2
sin^2 x - cos^2 y = -2/3

Similarly, for the second equation:
3(sin 2x - sin 2y) = 0
3(2sin x cos x - 2sin y cos y) = 0 (using the double angle formula)
2sin x cos x - 2sin y cos y = 0
sin x cos x - sin y cos y = 0
sin (x - y) = 0 (using the identity sin A cos B - cos A sin B = sin (A - B))

Now, we have two equations:
sin^2 x - cos^2 y = -2/3
sin (x - y) = 0

From the second equation, we have x - y = 0, which means x = y. Substituting this into the first equation, we have:
sin^2 x - cos^2 x = -2/3
2sin^2 x = -2/3
sin^2 x = -1/3

This is not possible, as the range of sine is from -1 to 1. Therefore, there is no solution for this problem.

2. For this problem, we can use the double angle formula for sine and cosine, as well as the sum and difference formula for cosine. We have:
sin^3 10 + cos^3 40 - sin^3 70
sin^3 10 + (cos 40)^3 - (sin