Question Statement

Solve the following trigonometric equations graphically:

(i) $cos θ = \frac{θ}{2}; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (ii) $tan θ = \frac{θ}{2}; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (iii) $sin θ = θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (iv) $tan θ = θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (v) $sin θ = 2 θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (vi) $cos θ = 2 θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (vii) $sin (\frac{θ}{2}) = θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (viii) $cos (\frac{θ}{2}) = θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (ix) $tan (\frac{θ}{2}) = θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (x) $cos (\frac{θ}{2}) = 2 θ; θ \in [- \frac{π}{2}, \frac{π}{2}]$ (xi) $sin θ = cos θ; θ \in [0, π]$ (xii) $sin θ = tan θ; θ \in [0, π]$

Background and Explanation

To solve an equation of the form $f (θ) = g (θ)$ graphically, we plot the curves $y = f (θ)$ and $y = g (θ)$ on the same set of axes. The solutions to the equation are the $θ$ -coordinates of the points where the two graphs intersect.

Solution

Part (i)

Equation: $cos θ = \frac{θ}{2}$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

To solve this, we plot $y = cos θ$ and $y = \frac{θ}{2}$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos θ$	$0$	$0.866$	$0.5$	$0$	$0.5$	$0.866$	$0$
$y = \frac{θ}{2}$	$- \frac{π}{4}$	$- \frac{π}{6}$	$- \frac{π}{12}$	$0$	$\frac{π}{12}$	$\frac{π}{6}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (ii)

Equation: $tan θ = \frac{θ}{2}$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan θ$ and $y = \frac{θ}{2}$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan θ$	$- \infty$	$- 1.732$	$- 0.577$	$0$	$0.577$	$1.732$	$\infty$
$y = \frac{θ}{2}$	$- \frac{π}{4}$	$- \frac{π}{6}$	$- \frac{π}{12}$	$0$	$\frac{π}{12}$	$\frac{π}{6}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (iii)

Equation: $sin θ = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin θ$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin θ$	$- 1$	$- 0.866$	$- 0.5$	$0$	$0.5$	$0.866$	$1$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (iv)

Equation: $tan θ = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan θ$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan θ$	$- \infty$	$- 1.732$	$- 0.577$	$0$	$0.577$	$1.732$	$\infty$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (v)

Equation: $sin θ = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin θ$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin θ$	$- 1$	$- 0.866$	$- 0.5$	$0$	$0.5$	$0.866$	$1$
$y = 2 θ$	$- π$	$- \frac{2 π}{3}$	$- \frac{π}{3}$	$0$	$\frac{π}{3}$	$\frac{2 π}{3}$	$π$

The solution is the point of intersection of the curves as shown in the graph.

Part (vi)

Equation: $cos θ = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos θ$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = 2 θ$	$- π$	$- \frac{2 π}{3}$	$- \frac{π}{3}$	$0$	$\frac{π}{3}$	$\frac{2 π}{3}$	$π$

The solution is the point of intersection of the curves as shown in the graph.

Part (vii)

Equation: $sin (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin \frac{θ}{2}$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin \frac{θ}{2}$	$- 0.707$	$- 0.5$	$- 0.259$	$0$	$0.259$	$0.5$	$0.707$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (viii)

Equation: $cos (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos \frac{θ}{2}$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos \frac{θ}{2}$	$0.707$	$0.866$	$0.97$	$1$	$0.97$	$0.866$	$0.707$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (ix)

Equation: $tan (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan (\frac{θ}{2})$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan (\frac{θ}{2})$	$- 1$	$- 0.577$	$- 0.27$	$0$	$0.27$	$0.577$	$1$
$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (x)

Equation: $cos (\frac{θ}{2}) = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos (\frac{θ}{2})$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos (\frac{θ}{2})$	$0.707$	$0.866$	$0.97$	$0$	$0.97$	$0.866$	$0.707$
$y = 2 θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (xi)

Equation: $sin θ = cos θ$ for $θ \in [0, π]$

We plot $y = sin θ$ and $y = cos θ$ .

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{5 π}{6}$	$π$
$y = sin θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = cos θ$	$1$	$0.866$	$0.5$	$0$	$- 0.5$	$- 0.866$	$- 1$

The solution is the point of intersection of the curves as shown in the graph.

Part (xii)

Equation: $sin θ = tan θ$ for $θ \in [0, π]$

We plot $y = sin θ$ and $y = tan θ$ .

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{5 π}{6}$	$π$
$y = sin θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = tan θ$	$0$	$0.577$	$1.732$	$\infty$	$- 1.732$	$- 0.5$	$0$

The solution is the point of intersection of the curves as shown in the graph.

Key Formulas or Methods Used

Graphical Method: Solving $f (θ) = g (θ)$ by finding the intersection of $y = f (θ)$ and $y = g (θ)$ .
Trigonometric Values: Standard values for $sin θ, cos θ,$ and $tan θ$ at multiples of $\frac{π}{6}$ and $\frac{π}{4}$ .
Linear Functions: Plotting lines of the form $y = m θ$ .

Summary of Steps

Identify the two functions $y_{1}$ and $y_{2}$ that make up the equation.
Create a table of values for both functions over the specified interval (e.g., $[- \frac{π}{2}, \frac{π}{2}]$ ).
Plot the points from the table on a coordinate system where the horizontal axis represents $θ$ .
Draw the curves/lines connecting the points.
Locate the intersection point(s); the $θ$ -value at these points is the solution to the equation.

Question Statement

Solve the following trigonometric equations graphically:

Background and Explanation

Solution

Part (i)

Equation: $cos θ = \frac{θ}{2}$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

To solve this, we plot $y = cos θ$ and $y = \frac{θ}{2}$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos θ$	$0$	$0.866$	$0.5$	$0$	$0.5$	$0.866$	$0$
$y = \frac{θ}{2}$	$- \frac{π}{4}$	$- \frac{π}{6}$	$- \frac{π}{12}$	$0$	$\frac{π}{12}$	$\frac{π}{6}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (ii)

Equation: $tan θ = \frac{θ}{2}$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan θ$ and $y = \frac{θ}{2}$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan θ$	$- \infty$	$- 1.732$	$- 0.577$	$0$	$0.577$	$1.732$	$\infty$
$y = \frac{θ}{2}$	$- \frac{π}{4}$	$- \frac{π}{6}$	$- \frac{π}{12}$	$0$	$\frac{π}{12}$	$\frac{π}{6}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (iii)

Equation: $sin θ = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin θ$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin θ$	$- 1$	$- 0.866$	$- 0.5$	$0$	$0.5$	$0.866$	$1$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (iv)

Equation: $tan θ = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan θ$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan θ$	$- \infty$	$- 1.732$	$- 0.577$	$0$	$0.577$	$1.732$	$\infty$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (v)

Equation: $sin θ = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin θ$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin θ$	$- 1$	$- 0.866$	$- 0.5$	$0$	$0.5$	$0.866$	$1$
$y = 2 θ$	$- π$	$- \frac{2 π}{3}$	$- \frac{π}{3}$	$0$	$\frac{π}{3}$	$\frac{2 π}{3}$	$π$

The solution is the point of intersection of the curves as shown in the graph.

Part (vi)

Equation: $cos θ = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos θ$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = 2 θ$	$- π$	$- \frac{2 π}{3}$	$- \frac{π}{3}$	$0$	$\frac{π}{3}$	$\frac{2 π}{3}$	$π$

The solution is the point of intersection of the curves as shown in the graph.

Part (vii)

Equation: $sin (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = sin \frac{θ}{2}$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = sin \frac{θ}{2}$	$- 0.707$	$- 0.5$	$- 0.259$	$0$	$0.259$	$0.5$	$0.707$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (viii)

Equation: $cos (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos \frac{θ}{2}$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos \frac{θ}{2}$	$0.707$	$0.866$	$0.97$	$1$	$0.97$	$0.866$	$0.707$
$y = θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (ix)

Equation: $tan (\frac{θ}{2}) = θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = tan (\frac{θ}{2})$ and $y = θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = tan (\frac{θ}{2})$	$- 1$	$- 0.577$	$- 0.27$	$0$	$0.27$	$0.577$	$1$
$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (x)

Equation: $cos (\frac{θ}{2}) = 2 θ$ for $θ \in [- \frac{π}{2}, \frac{π}{2}]$

We plot $y = cos (\frac{θ}{2})$ and $y = 2 θ$ .

$θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$
$y = cos (\frac{θ}{2})$	$0.707$	$0.866$	$0.97$	$0$	$0.97$	$0.866$	$0.707$
$y = 2 θ$	$- \frac{π}{2}$	$- \frac{π}{3}$	$- \frac{π}{6}$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$

The solution is the point of intersection of the curves as shown in the graph.

Part (xi)

Equation: $sin θ = cos θ$ for $θ \in [0, π]$

We plot $y = sin θ$ and $y = cos θ$ .

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{5 π}{6}$	$π$
$y = sin θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = cos θ$	$1$	$0.866$	$0.5$	$0$	$- 0.5$	$- 0.866$	$- 1$

The solution is the point of intersection of the curves as shown in the graph.

Part (xii)

Equation: $sin θ = tan θ$ for $θ \in [0, π]$

We plot $y = sin θ$ and $y = tan θ$ .

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{5 π}{6}$	$π$
$y = sin θ$	$0$	$0.5$	$0.866$	$1$	$0.866$	$0.5$	$0$
$y = tan θ$	$0$	$0.577$	$1.732$	$\infty$	$- 1.732$	$- 0.5$	$0$

The solution is the point of intersection of the curves as shown in the graph.

Key Formulas or Methods Used

Graphical Method: Solving $f (θ) = g (θ)$ by finding the intersection of $y = f (θ)$ and $y = g (θ)$ .
Trigonometric Values: Standard values for $sin θ, cos θ,$ and $tan θ$ at multiples of $\frac{π}{6}$ and $\frac{π}{4}$ .
Linear Functions: Plotting lines of the form $y = m θ$ .

Summary of Steps

Identify the two functions $y_{1}$ and $y_{2}$ that make up the equation.
Create a table of values for both functions over the specified interval (e.g., $[- \frac{π}{2}, \frac{π}{2}]$ ).
Plot the points from the table on a coordinate system where the horizontal axis represents $θ$ .
Draw the curves/lines connecting the points.
Locate the intersection point(s); the $θ$ -value at these points is the solution to the equation.

9.2 Q-1

Question Statement

Background and Explanation

Solution

Part (i)

Part (ii)

Part (iii)

Part (iv)

Part (v)

Part (vi)

Part (vii)

Part (viii)

Part (ix)

Part (x)

Part (xi)

Part (xii)

Key Formulas or Methods Used

Summary of Steps

9.2 Q-1

Question Statement

Background and Explanation

Solution

Part (i)

Part (ii)

Part (iii)

Part (iv)

Part (v)

Part (vi)

Part (vii)

Part (viii)

Part (ix)

Part (x)

Part (xi)

Part (xii)

Key Formulas or Methods Used

Summary of Steps