if alpha and beta are 2 different values of θ lying between 0 and 2π which satisfy the equation 6cosθ+8 sinθ=9 find the value of sin alpha + beta
0.6 cosè + 0.8 sinè = 0.9
Let sinx = 0.6; then
cosx = 0.8 and x = 36.87 degrees
sinx cosè + cosx sinè = 0.9
sin (x + è) = 0.9
x + è = 64.16 degrees or 115.84 degrees
alpha = è1 = 27.29 degrees
beta = è2 = 78.97
alpha + beta = 106.26 degrees
sin (alpha + beta) = 0.9600
drwls,
that is amazingly clever.
thank you very much. can u solve the same by quadratic equation method
yes, it can be done that way, but I'd rather leave that to you.
Write it as
6cosθ= 9 - 8 sinθ
Square both sides and the left side can be written 36(1 - sin^2 θ)
Then you have a quadratic in sin theta to solve. Have fun.
To find the value of sin(α + β), we first need to find the values of sin(α) and sin(β). Then, we can add those values to get sin(α + β).
Given the equation:
6cos(θ) + 8sin(θ) = 9
Let's rewrite this equation in terms of sin(θ) by dividing both sides by 10 to simplify:
(6/10)cos(θ) + (8/10)sin(θ) = 9/10
Now, notice that this equation is in the form of sin(θ - φ) = a, where a is a constant. We can use the identity sin(θ - φ) = sin(θ)cos(φ) - cos(θ)sin(φ) to rewrite the equation as follows:
sin(θ - φ) = 9/10
Comparing this to the identity, we can see that sin(φ) = 6/10 and cos(φ) = 8/10.
Now, we need to find the values of α and β that satisfy sin(θ - φ) = 9/10.
Since α and β are two different values of θ that satisfy the equation, we have two different angles at which sin(θ - φ) is equal to 9/10.
We can use the inverse trigonometric functions (arcsin) to find α and β.
For the first value of θ, let's say α:
θ - φ = arcsin(9/10)
For the second value of θ, let's say β:
θ - φ = π - arcsin(9/10)
Now, we need to solve for θ in both equations.
For α:
θ = φ + arcsin(9/10)
For β:
θ = φ + π - arcsin(9/10)
Finally, we can find sin(α) and sin(β) by substituting the values of θ in the equations:
sin(α) = sin(φ + arcsin(9/10))
sin(β) = sin(φ + π - arcsin(9/10))
Once we have these values, we can add them together to find sin(α + β):
sin(α + β) = sin(α) + sin(β)