Consider the equation 7sin^2x - (4sin2^x)/cosx = -1
a) Put the equation in standard quadratic trig equation form.
Why is it 7sin^2x - 8sinx + 1 =0?
To put the equation in standard quadratic trig equation form, we need to rewrite it using the trigonometric identity for double angles.
First, let's rewrite the term (4sin^2x)/cosx as (4sin2^x)/cosx.
Using the double angle identity for sine, sin2x = 2sinxcosx, we can rewrite the term as (8sinxcosx)/cosx.
Now, let's simplify the equation further:
7sin^2x - (8sinxcosx)/cosx = -1
Since cosx/cosx equals 1, we can simplify the equation to:
7sin^2x - 8sinx = -1
To put the equation in standard quadratic trig equation form, we move all terms to one side of the equation:
7sin^2x - 8sinx + 1 = 0
Now, the equation is in standard quadratic trig equation form.
The reason it becomes 7sin^2x - 8sinx + 1 = 0 is because we simplified the term (4sin2^x)/cosx using the double angle identity for sine, and then rearranged the equation to have all terms on one side.
To put the equation in standard quadratic trigonometric equation form, we need to simplify the terms and combine like terms.
Given: 7sin^2x - (4sin2^x)/cosx = -1.
First, let's simplify the second term: (4sin2^x)/cosx.
Since sin2^x can be written as sin^2x, we can rewrite the term as (4sin^2x)/cosx.
Now, we have: 7sin^2x - (4sin^2x)/cosx = -1.
To combine the terms, we need a common denominator for sin^2x and (sin^2x)/cosx, which is cosx.
Multiply the first term by cosx to get: 7sin^2x * cosx.
Now, the equation becomes: 7sin^2x * cosx - (4sin^2x)/cosx = -1.
Next, let's simplify by multiplying through by cosx:
7sin^2x * cos^2x - 4sin^2x = -cosx.
Now, distribute the sin^2x:
7sin^2x * cos^2x - 4sin^2x = -cosx.
Finally, rearrange the terms to get the equation in standard quadratic trigonometric equation form:
7sin^2x * cos^2x - 4sin^2x + cosx = 0.
So, the equation in standard quadratic trigonometric equation form is 7sin^2x - 8sinx + 1 = 0.