Let's start by recalling the Negative Angle Identity and the Cofunction Identity that we will use.
Cofunction Identity
tan (π/2-θ)=cot θ [0.8em]
Negative Angle Identity
tan (- θ)=-tan θ
To solve the given equation, we will first rewrite it using these identities.
Finally, we need to check if there are any other possible angles that satisfy this equation within the given range.
Given Range: 0 ≤ θ < 2π
Recall that the tangent of an angle in standard positionintersects the unit circle at the point (x,y). Let's plot all the points on the unit circle where the ratio yx equals 1.
As we can see above, the angles whose tangent is 1 are π4 and 5π4.
tan θ = -1
Next, we will isolate θ.
tan θ =-1 ⇔ θ =tan ^(- 1) (-1)
Let's use a calculator to find one value for θ.
Finally, we need to check if there are any other possible angles that satisfy this equation within the given range.
Given Range: 0 ≤ θ < 2π
Recall that the tangent of an angle in standard position intersects the unit circle at the point (x,y). Let's plot all the points on the unit circle where the ratio yx equals -1.
As we can see above, the angles whose tangent is -1 are 3π4 and 7π4. Therefore, the solutions for the equation in the given range are π4, 5π4, 3π4, and 7π4.
