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Can someone please help me with b?

Can someone please help me with b?-example-1
User Gauls
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1 Answer

4 votes

Answer:

x = -π/2, π/6, 5π/6

Explanation:

You are given a quadratic equation in sin(x) and cos(x) and asked for the values of x that make it true. The domain of interest is given.

Trig identity

The trig identity ...

cos(x)² = 1 -sin(x)²

is useful for turning this equation into a quadratic in sin(x). Making use of it, the equation becomes ...

1 -sin(x)² -sin(x)² = sin(x)

Solution

Subtracting the left-side expression from both sides gives the standard-form equation ...

2sin(x)² +sin(x) -1 = 0

Factoring this equation, we have ...

(2sin(x) -1)(sin(x) +1) = 0

The values of x that make these factors zero are ...

2sin(x) -1 = 0 ⇒ sin(x) = 1/2 ⇒ x = π/6 or 5π/6

sin(x) +1 = 0 ⇒ sin(x) = -1 ⇒ x = -π/2

__

Additional comment

Sometimes the quadratic nature of the equation is easier to see if you make a substitution for the trig function.

sin(x) = z ⇒ 2z² +z -1 = 0 ⇒ (2z -1)(z +1) = 0 ⇒ z = 1/2 or -1

At this point, you need to find the values of x that correspond to these values of z. (In short, you're not quite done yet when you find the solutions for z.)

Can someone please help me with b?-example-1
User Rolanda
by
7.5k points

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