Solve the equation:

[tex]\[ p = s_3^{110} \frac{k}{km} \][/tex]

(i) Given:

- The area of a triangle is [tex]\( kp \)[/tex].
- The shortest distance from [tex]\( k \)[/tex] to [tex]\( sp \)[/tex].

Solve the equation:

[tex]\[ \tan Q = 0.7 \][/tex]

for:

[tex]\[ 0^\circ \leqslant Q \leqslant 180^\circ \][/tex]



Answer :

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].