To solve the given equation[tex]\(\ tan(Q) = 0.7 \ \)[/tex] for [tex]\( \ 0^\circ \leq Q \leq 180^\circ \)[/tex]:
### Step-by-Step Solution:
1. Identify the range and properties of the tangent function:
- Given: [tex]\(\ tan(Q) = 0.7 \)[/tex]
- We need to find all angles [tex]\( Q \)[/tex] such that [tex]\( 0^\circ \leq Q \leq 180^\circ \)[/tex].
- The tangent function [tex]\( \ tan(Q) \)[/tex] is positive in the first and third quadrants, but for our range [tex]\(0^\circ\)[/tex] to [tex]\(180^\circ\)[/tex], we will consider the first and second quadrants.
2. Find the Principal Value:
- Use an inverse tangent function to determine the principal value [tex]\( Q \)[/tex].
- [tex]\( Q = \ tan^{-1}(0.7) \)[/tex] gives us one solution. We convert this to degrees.
Using calculations, we get:
[tex]\[
Q \approx 34.99202019855866^\circ
\][/tex]
3. Determine the Solutions within the Given Range:
- In the first quadrant (where tangent is positive), the principal value [tex]\( 34.99202019855866^\circ \)[/tex] is one solution.
- In the second quadrant, the tangent of an angle still results in a positive value. To find the second angle, subtract the principal value from [tex]\( 180^\circ \)[/tex]:
[tex]\[
Q_{\text{second quadrant}} = 180^\circ - 34.99202019855866^\circ = 145.00797980144134^\circ
\][/tex]
4. Conclusion:
- The solutions to the equation [tex]\(\ tan(Q) = 0.7 \)[/tex] in the interval [tex]\( 0^\circ \leq Q \leq 180^\circ \)[/tex] are:
[tex]\[
Q \approx 34.99202019855866^\circ \quad \text{and} \quad Q \approx 145.00797980144134^\circ
\][/tex]
Therefore, the angles [tex]\( Q \)[/tex] satisfying the given equation within the specified range are approximately [tex]\( 34.99202019855866^\circ \)[/tex] and [tex]\( 145.00797980144134^\circ \)[/tex].
