Let's start by recalling the Cofunction Identity for sine and the Negative Angle Identity for sine.
Cofunction Identity
sin (π/2-θ)=cos θ [0.8em]
Negative Angle Identity
sin (- θ)=- sin θ
To solve the given equation, we will first rewrite it using these identities.
Now, let's recall the Tangent Identity. Note that we will need to rearrange the terms so that we can use them for our expression.
tan θ = sin θ/cos θ ⇔ sin θ/cos θ = tan θ
We can substitute sin θcos θ for tan θ in our expression.
Now, we need to convert this angle in terms of π. To do this, we can multiply the angle by π3.141592....
θ =- 1.107148...(π/3.141592...)
Let's simplify the above expression.
We can see that the obtained solution is not in the given range!
Given Range: 0 ≤ θ < 2π
Recall that if an angle is negative, it is being measured clockwise. Therefore, to find its corresponding positive value, we need to add 2π.
Finally, we need to check if there are any other possible angles that satisfy this equation within the given range. Recall that the tangent is negative in the second and fourth quadrants. Therefore, for symmetry reasons, we will subtract 7π20 from π.
As we can see above, the angles whose tangent is - 2 are 33π20 and 13π20. These are the solutions for the equation in the given range.
