Angle between Lines

If the acute angle \( \theta \) between two straight lines \( y = m_1 x + a \) and \( y = m_2 x + b \), then
$$ \large \tan{\theta} = \left|\frac{m_1-m_2}{1+m_1 \times m_2}\right|$$
Example 1: Straight Lines in Standard Form
Find the acute angles between the lines \( y = -2x + 4 \) and \( y = 3x + 2 \), giving your answer to the nearest degree.
\( \begin{aligned} \displaystyle
m_1 &= -2 \text{ and } m_2 = 3 \\
\tan \theta &= \left| \frac{-2-3}{1+(-2) \times 3}\right| \\
&= 1 \\
\theta &= \tan^{-1}1 \\
\therefore \theta &= 45^{\circ}
\end{aligned} \)
Example 2: Straight Lines in General Form
Find the acute angles between the lines \( 2y = x + 1 \) and \( 2x-3y = 4 \), giving your answer to the nearest degree.
\( \begin{aligned} \displaystyle
2y &= x + 1 \\
y &= \frac{1}{2}x + \frac{1}{2} \\
m_1 &= \frac{1}{2} \\
2x-3y &= 4 \\
3y &= 2x + 4 \\
y &= \frac{2}{3}x + \frac{3}{4} \\
m_2 &= \frac{2}{3} \\
\tan{\theta} &= \left|\frac{\dfrac{1}{2}-\dfrac{2}{3}}{1+\dfrac{1}{2} \times \dfrac{2}{3}}\right| \\
&= \frac{1}{8} \\
\theta &= \tan^{-1} \frac{1}{8} \\
\therefore \theta &= 7^{\circ}
\end{aligned} \)
Example 3: Angle between Lines to the Nearest Minute
Find the acute angle between the lines \( 2x+y+1=0 \) and \( x+y+4=0 \), correcting to the nearest minute.
\( \begin{aligned} \displaystyle
2x+y+1 &= 0 \\
y &= -2x-1 \\
m_1 &= -2 \\
x+y+4 &= 0 \\
y &= -x-1 \\
m_2 &= -1 \\
\tan{\theta} &= \left|\frac{-2-(-1)}{1+(-2) \times (-1)}\right| \\
&= \frac{1}{3} \\
\theta &= \tan^{-1}\frac{1}{3} \\
\therefore \theta &= 18^{\circ}26^{\prime}
\end{aligned} \)
Example 4: Two Points and a Straight Line
Find the acute angle between the line \( 2x-5y+1=0 \) and the line joining \( (-1,2) \) and \( (5,3) \), correcting to the nearest minute.
\( \begin{aligned} \displaystyle
2x-5y+1 &= 0 \\
5y &= 2x+1 \\
y &= \frac{2}{5}x + \frac{1}{5} \\
m_1 &= \frac{2}{5} \\
m_2 &= \frac{3-2}{5-(-1)} \\
&= \frac{1}{6} \\
\tan{\theta} &= \left|\frac{\dfrac{2}{5}-\dfrac{1}{6}}{1+\dfrac{2}{5} \times \dfrac{1}{6}}\right| \\
&= \frac{7}{32} \\
\theta &= \tan^{-1}\frac{7}{32} \\
\therefore \theta &= 12^{\circ}20^{\prime}
\end{aligned} \)
Example 5: Angles at Intersection of Exponential Graphs
Find the angle between the tangents drawn to the curve \( y_1 = e^{x} \) and \( y_2 = e^{-x} \) at their point of intersection.
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
e^{x} &= e^{-x} &\color{red} \text{find the intersection} \\
e^{x} \times e^{x} &= e^{-x} \times e^{x} \\
e^{2x} &= 1 \\ &= e^0 \\
2x &= 0 \\
x &= 0 \\
\frac{dy_1}{dx} &= e^{x} \\
m_1 &= \left|\frac{dy_1}{dx}\right|_{x=0} \\
&= e^{0} \\
&= 1 \\
\frac{dy_2}{dx} &= -e^{x} \\
m_2 &= \left|\frac{dy_2}{dx}\right|_{x=0} \\
&= -e^{0} \\
&= -1 \\
\tan{\theta} &= \left|\frac{1+1}{1-1}\right| \\
&= \left|\frac{1+1}{0}\right| \\
&= \text{undefined} \\
\therefore \theta &= 90^{\circ}
\end{aligned} \)
Example 6: Angles at Intersection of Trigonometric Graphs
Find the acute angle between the curves \( y_1 = \sin{x} \) and \( y_2 = \sin{2x} \), at their point of intersection, where \(0 \lt x \lt 180^{\circ} \), correcting to the neatest degree.
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\sin{2x} &= \sin{x} \\
2\sin{x}\cos{x} &= \sin{x} \\
\sin{x}(2\cos{x}-1) &= 1 \\
\sin{x} &= 0 \text{ or } 2\cos{x}-1 = 0 \\
\sin{x} &\ne 0 &\color{red} 0 \lt x \lt 180^{\circ} \\
\cos{x} &= \frac{1}{2} \\
x &= 60^{\circ} \\
\frac{dy_1}{dx} &= \cos{x} \\
\left.\frac{dy_1}{dx} \right|_{x = 60^{\circ}} &= \cos{60^{\circ}} &\color{red} \\
m_1 &= \frac{1}{2} \color{red} \cdots (1) \\
\frac{dy_2}{dx} &= 2\cos{2x} \\
\left.\frac{dy_2}{dx}\right|_{x=60^{\circ}} &= 2\cos{120^{\circ}} \\
&= -1 \\
m_2 &= -1 \color{red} \cdots (2) \\
\tan\theta &= \left|\frac{\dfrac{1}{2}-(-1)}{1+\dfrac{1}{2} \times (-1)}\right| &\color{red} \text{by (1) and (2)} \\
&= 3 \\
\theta &= \tan^{-1}{3} \\
&= 72^{\circ}
\end{aligned} \)
Example 7: Finding Gradients using Angle between Straight Lines
The acute angle between the lines \( 2x-y-7=0 \) and \( y=mx+3 \) is \( 25^{\circ} \), find the value(s) of (m), correct to one decimal place.
\( \begin{aligned} \displaystyle \require{AMSsymbols} \require{color}
\left|\frac{2-m}{1+2m}\right| &= \tan{25^{\circ}} \\
\frac{2-m}{1+2m} = \tan{25^{\circ}} &\text{ or } \frac{2-m}{1+2m} = -\tan{25^{\circ}} \\
\frac{2-m}{1+2m} &= \tan{25^{\circ}} \\
2-m &= \tan{25^{\circ}} + 2m\tan{25^{\circ}} \\
-m-2m\tan{25^{\circ}} &= \tan{25^{\circ}}-2 \\
m + 2m\tan{25^{\circ}} &= 2-\tan{25^{\circ}} \\
m(1 + 2\tan{25^{\circ}}) &= 2-\tan{25^{\circ}} \\
m &= \frac{2-\tan{25^{\circ}}}{1 + 2\tan{25^{\circ}}} \\
m &= 0.8 \color{red} \cdots (1) \\
\frac{2-m}{1+2m} &= -\tan{25^{\circ}} \\
2-m &= -\tan{25^{\circ}}-2m\tan{25^{\circ}} \\
-m + 2m\tan{25^{\circ}} &= -\tan{25^{\circ}}-2 \\
2m\tan{25^{\circ}}-m &= \tan{25^{\circ}} + 2 \\
m(2\tan{25^{\circ}}-1) &= \tan{25^{\circ}} + 2 \\
m &= \frac{\tan{25^{\circ}} + 2}{2\tan{25^{\circ}}-1} \\
m &= 36.6 \color{red} \cdots (2) \\
\therefore m &= 0.8 \text{ or } m = 36.6 &\color{red} \text{by (1) and (2)}
\end{aligned} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume