In ∆PQR, if 3sinP + 4cosQ = 1 and 4sinQ + 3cosP = 1, we need to find the value of angle R.

Given:
- 3sinP + 4cosQ = 1
- 4sinQ + 3cosP = 1

To find the value of angle R, we can use the trigonometric identities of sine and cosine.

Using Trigonometric Identities:
1. Start by rearranging the first equation: 3sinP = 1 - 4cosQ
2. Square both sides of the equation: (3sinP)^2 = (1 - 4cosQ)^2
3. Expand the equation: 9sin^2P = 1 - 8cosQ + 16cos^2Q
4. Rewrite sin^2P as 1 - cos^2P using the Pythagorean identity: 9(1 - cos^2P) = 1 - 8cosQ + 16cos^2Q
5. Simplify the equation: 9 - 9cos^2P = 1 - 8cosQ + 16cos^2Q
6. Rearrange the equation: 9cos^2P + 16cos^2Q - 8cosQ - 9 = 0

Quadratic Equation:
7. Treat cosP as x and cosQ as y to convert the equation into a quadratic form: 9x^2 + 16y^2 - 8y - 9 = 0

Factorization:
8. We need to factorize the quadratic equation: (3x - 1)(3x + 9) + (4y - 3)(4y + 3) = 0
9. Simplify the equation: (3x - 1)(3x + 9) + 4(4y - 3)(y + 1) = 0

Equating Factors:
10. Equate each factor to zero:
- 3x - 1 = 0, which gives x = 1/3
- 3x + 9 = 0, which gives x = -3
- 4y - 3 = 0, which gives y = 3/4
- 4y + 3 = 0, which gives y = -3/4

Substituting Back:
11. Substitute the values of x and y back into the original equations to find the corresponding values of P and Q:
- 3sinP + 4cosQ = 1
* When x = 1/3 and y = 3/4: 3sinP + 4(cosQ) = 1
* Substitute sinP = 1 - 4(cosQ)/3: 3(1 - 4(cosQ)/3) + 4(cosQ) = 1
* Simplify the equation: 1 - 4(cosQ) + 4(cosQ) = 1
* The equation becomes: 1 = 1, which holds true.

- 4sinQ