For example you have an equation as such 5sinx-3cosx=1 ..How do you work it out
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5sinx - 3cosx = 1
5^2+3^2 = 34, so that is the same as
√34 (5/√34 sinx - 3/√34 cosx) = 1
Now, if y is an angle such that tany = 3/5, you have
√34 (cosy sinx - siny cosx) = 1
sin(x-y) = 1/√34
x-y = 9.875°
and, since y = 30.964°, that means
x - 30.964 = 9.875
x = 40.839° or 201.084° + 2kπ
To solve the equation 5sinx - 3cosx = 1, you can follow these steps:
Step 1: Rearrange the equation to isolate either sinx or cosx. Since sinx and cosx are related by the Pythagorean identity sin^2x + cos^2x = 1, we can square both sides of the equation and simplify:
(5sinx)^2 + (-3cosx)^2 = 1^2
25sin^2x + 9cos^2x = 1
Step 2: Use the Pythagorean identity sin^2x + cos^2x = 1 to rewrite the equation:
25sin^2x + 9(1 - sin^2x) = 1
25sin^2x + 9 - 9sin^2x = 1
16sin^2x = 8
sin^2x = 8/16
sin^2x = 1/2
Step 3: Take the square root of both sides to find the value of sinx:
sinx = ±√(1/2)
Step 4: Find the values of sinx. Since sinx is positive in quadrants I and II, and negative in quadrants III and IV, we have:
sinx = √(1/2) or sinx = -√(1/2)
Step 5: Use the values of sinx to find the values of cosx. Recall that cosx = ±√(1 - sin^2x):
For sinx = √(1/2):
cosx = ±√(1 - (1/2))
cosx = ±√(1/2)
For sinx = -√(1/2):
cosx = ±√(1 - (1/2))
cosx = ±√(1/2)
So the solutions for the equation 5sinx - 3cosx = 1 are:
x = arcsin(√(1/2)) + 2nπ, x = arcsin(-√(1/2)) + 2nπ, x = arccos(√(1/2)) + 2nπ, and x = arccos(-√(1/2)) + 2nπ, where n is an integer.
To solve the equation 5sinx - 3cosx = 1, we will use trigonometric identities to transform it into a more manageable form. Here's how you can work it out step-by-step:
Step 1: Rearrange the equation
Move the 1 to the left side of the equation to obtain:
5sinx - 3cosx - 1 = 0
Step 2: Use the identity sin^2(x) + cos^2(x) = 1
We can rewrite the equation using this identity by dividing each term by √(5^2 + (-3)^2), which is the square root of (25 + 9) = √34.
(sin(x)/√34)^2 + (cos(x)/√34)^2 - 1/√34 = 0
Step 3: Simplify the equation
Let's define a new variable p = sin(x)/√34 and q = cos(x)/√34, so our equation becomes:
p^2 + q^2 - 1/√34 = 0
Step 4: Apply the Pythagorean identity
We can now apply the Pythagorean identity p^2 + q^2 = 1, which states that the sum of the squares of sine and cosine is always equal to 1. Substituting this into the equation, we get:
1 - 1/√34 = 0
We can further simplify this expression to:
√34 - 1 = 0
Step 5: Solve for x
To solve for x, we need to find the values of sin(x) and cos(x). Since the equation √34 - 1 = 0 is false, it means that there is no solution for x that satisfies the original equation 5sinx - 3cosx = 1.
In summary, there is no solution for the equation 5sinx - 3cosx = 1. We arrived at this conclusion by manipulating the equation using trigonometric identities and determining that the resulting equation has no valid solutions.