Solving Trigonometric Equations Revision Notes for Grade 10 NSC Matric Mathematics

Solving Trigonometric Equations

Introduction

Solving trigonometric equations involves finding unknown lengths and angles in right-angled triangles, as well as solving more general trigonometric equations. This process uses the fundamental trigonometric ratios and their inverse functions to determine missing values.

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The approach varies depending on what you need to find:

Finding lengths: Use trigonometric ratios to create equations
Finding angles: Use inverse trigonometric functions
Solving equations: Rearrange and apply inverse functions

Finding unknown lengths in right-angled triangles

When you need to find an unknown side length in a right-angled triangle, you can use the trigonometric ratios if you know one angle and one side length.

The trigonometric ratios

The three main trigonometric ratios are:

Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Remember that the hypotenuse is always the longest side opposite the right angle. The opposite side is across from your angle of interest, and the adjacent side is next to your angle of interest.

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Remember SOH CAH TOA:

Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent

This mnemonic helps you remember which sides to use for each ratio.

Step-by-step method for finding lengths

Step 1: Identify the sides

Label the hypotenuse, opposite, and adjacent sides relative to the given angle
Determine which trigonometric ratio to use based on what you know and what you need to find

Step 2: Set up the equation

Write the appropriate trigonometric ratio
Substitute the known values

Step 3: Rearrange to solve for the unknown

Use algebra to isolate the unknown variable
Multiply or divide both sides as needed

Step 4: Calculate the answer

Use your calculator to find the numerical value
Round to the required number of decimal places

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Worked Example: Finding a length using sine

To find the length $x$ in this triangle:

Step 1: Identify the ratio needed

We have the hypotenuse (100) and need the opposite side ( $x$ )
Use: $\sin 50° = \frac{x}{100}$

Step 2: Rearrange the equation

$\sin 50° \times 100 = x$
$x = 100 \sin 50°$

Step 3: Calculate

$x = 100 \times 0.766...$
$x ≈ 76.60$

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Worked Example: Finding a length using cosine

For this triangle where we need to find $x$ :

Step 1: Identify the ratio needed

We have the hypotenuse (7) and need the adjacent side ( $x$ )
Use: $\cos 25° = \frac{7}{x}$

Step 2: Rearrange the equation

$\cos 25° \times x = 7$
$x = \frac{7}{\cos 25°}$

Step 3: Calculate

$x = \frac{7}{0.906...}$
$x ≈ 7.72$

Finding multiple unknown lengths

When you need to find two unknowns, use two different trigonometric ratios:

For the 25° angle:

$\tan 25° = \frac{x}{7}$ , so $x = 7 \times \tan 25°$
$\cos 25° = \frac{7}{y}$ , so $y = \frac{7}{\cos 25°}$

Calculating:

$x = 7 \times 0.466... ≈ 3.26$
$y = \frac{7}{0.906...} ≈ 7.72$

Finding unknown angles in right-angled triangles

When you know two side lengths and need to find an angle, you use the inverse trigonometric functions on your calculator.

Using inverse functions

The inverse trigonometric functions work backwards from the ratios to find angles:

$\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$
$\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$
$\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

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Calculator Technique for Inverse Functions:

To use inverse functions on your calculator:

Press SHIFT (or 2nd) to access inverse functions
Press the trigonometric function button (sin, cos, or tan)
Enter the ratio value
Press = to get the angle

Always remember: SHIFT key is essential for finding angles!

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Worked Example: Finding an angle

To find angle $\theta$ in this triangle:

Step 1: Identify the ratio

We have opposite (50) and adjacent (100)
Use: $\tan \theta = \frac{50}{100} = 0.5$

Step 2: Use inverse function

Press: SHIFT → tan → 0.5 → =
Result: $\theta ≈ 26.6°$

Solving general trigonometric equations

Beyond right-angled triangles, you may need to solve trigonometric equations where the triangle isn't shown.

Method for solving trigonometric equations

Step 1: Rearrange the equation

Isolate the trigonometric function on one side
Ensure the value is between -1 and 1 for sine and cosine

Step 2: Use the inverse function

Apply the appropriate inverse trigonometric function
Use your calculator to find the angle

Step 3: Check for valid solutions

Remember that sine and cosine have maximum values of 1
If the required value is greater than 1, there is no solution

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Worked Example: Basic equation

Find $\theta$ if $\cos \theta = 0.2$

Step 1: The equation is already arranged

$\cos \theta = 0.2$

Step 2: Use inverse cosine

Press: SHIFT → cos → 0.2 → =
Result: $\theta ≈ 78.46°$

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Worked Example: Equation with coefficient

Find $\theta$ if $3\sin \theta = 2.4$

Step 1: Rearrange

$3\sin \theta = 2.4$
$\sin \theta = \frac{2.4}{3} = 0.8$

Step 2: Use inverse sine

Press: SHIFT → sin → 0.8 → =
Result: $\theta ≈ 53.13°$

Cases with no solution

Sometimes trigonometric equations have no solution. This occurs when:

The required value for sine or cosine is greater than 1
The required value for sine or cosine is less than -1

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Recognizing No Solution Cases:

Example: Solve $3\sec \alpha = 1.4$

Since $\sec \alpha = \frac{1}{\cos \alpha}$ :

$\frac{3}{\cos \alpha} = 1.4$
$\cos \alpha = \frac{3}{1.4} ≈ 2.14$

Since 2.14 > 1 and the maximum value of cosine is 1, there is no solution.

Always check if your calculated ratio exceeds the valid range for trigonometric functions!

Important calculator notes

Always use the SHIFT key to access inverse functions
Round answers appropriately (usually to 1 decimal place for angles)
Check that your calculator is in degree mode
Write "no solution" clearly when equations cannot be solved

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Key Points to Remember:

The trigonometric ratios are: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
Use inverse functions (with SHIFT key) only when finding angles from known ratios
Maximum values for sine and cosine functions are 1 - values greater than 1 indicate no solution exists
Always identify triangle parts relative to the angle you're working with - the hypotenuse never changes, but opposite and adjacent depend on your reference angle
Calculator technique is crucial - practice using SHIFT for inverse functions and ensure degree mode is selected

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Common Exam Tips:

Identify the triangle parts carefully - label opposite, adjacent, and hypotenuse relative to the given angle
Choose the correct ratio based on which sides you know and which you need to find
Use inverse functions when finding angles, not when finding lengths
Check your calculator mode - ensure it's set to degrees
Round appropriately - typically 2 decimal places for lengths, 1 for angles
Recognize no-solution cases - when sine or cosine values exceed 1

Solving Trigonometric Equations (Grade 10 NSC Matric Mathematics): Revision Notes