Trignometry
Practice MCQsTrigonometry is an important branch of mathematics that studies the relationship between the angles and sides of a triangle. It is widely used in geometry, heights and distances, navigation, construction, engineering & physics.
Trigonometry is an important branch of mathematics that studies the relationship between angles and sides of triangles. It covers angle measurement in degrees and radians, trigonometric ratios, standard angle values, identities, sum and difference formulae, multiple and sub-multiple angles, inverse trigonometric functions, use of trigonometric tables, properties of triangles, and applications such as heights and distances.
What is Trigonometry?
Trigonometry mainly deals with ratios of sides of a right-angled triangle. These ratios are called trigonometric ratios. The most commonly used ratios are sine, cosine, and tangent.
In a right-angled triangle, if one acute angle is represented by \(\theta\), then the sides are identified as perpendicular, base, and hypotenuse. Trigonometric ratios help us find unknown sides or angles when some information is given.
| Ratio | Meaning | Formula |
|---|---|---|
| Sine | Ratio of perpendicular to hypotenuse. | \(\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}\) |
| Cosine | Ratio of base to hypotenuse. | \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}\) |
| Tangent | Ratio of perpendicular to base. | \(\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}\) |
| Cosecant | Reciprocal of sine. | \(\csc \theta = \frac{1}{\sin \theta}\) |
| Secant | Reciprocal of cosine. | \(\sec \theta = \frac{1}{\cos \theta}\) |
| Cotangent | Reciprocal of tangent. | \(\cot \theta = \frac{1}{\tan \theta}\) |
“Trigonometry converts angles and sides into useful mathematical relationships.”
Key points
- Understand degree and radian measures.
- Learn sine, cosine, and tangent clearly.
- Memorize standard angle values.
- Use \(\csc\) for cosecant in MathJax formulas.
- Practice simple trigonometric identities.
- Learn sum, difference, double and half-angle formulae.
- Understand inverse trigonometric functions.
- Use trigonometric tables where required.
- Apply ratios in heights and distances.
Angles and Their Measures: Degrees and Radians
Angles can be measured in degrees or radians. Competitive exam questions commonly use degrees, while higher mathematics often uses radians.
| Measure | Meaning | Important Conversion |
|---|---|---|
| Degree | A full circle is divided into \(360^\circ\). | \(180^\circ=\pi\) radians |
| Radian | Angle subtended at the centre by an arc equal to radius. | \(\pi\) radians \(=180^\circ\) |
| Degree to Radian | Multiply by \(\frac{\pi}{180}\). | \(60^\circ=60\times \frac{\pi}{180}=\frac{\pi}{3}\) |
| Radian to Degree | Multiply by \(\frac{180}{\pi}\). | \(\frac{\pi}{4}=\frac{\pi}{4}\times \frac{180}{\pi}=45^\circ\) |
Basic Parts of a Right-Angled Triangle
Trigonometric ratios are defined with respect to an acute angle \(\theta\) in a right-angled triangle. Before using formulas, identify the three sides correctly.
Hypotenuse
The side opposite to the right angle. It is always the longest side of the right-angled triangle.
Perpendicular
The side opposite to the angle \(\theta\). It is also called the opposite side.
Base
The side adjacent to the angle \(\theta\), excluding the hypotenuse. It is also called the adjacent side.
Formula and Identity Bank
\(\cos \theta = \frac{B}{H}\)
\(\tan \theta = \frac{P}{B}\)
\(\sec \theta = \frac{H}{B}\)
\(\cot \theta = \frac{B}{P}\)
\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
\(1+\tan^2 \theta = \sec^2 \theta\)
\(1+\cot^2 \theta = \csc^2 \theta\)
\(\cos(90^\circ-\theta)=\sin \theta\)
\(\tan(90^\circ-\theta)=\cot \theta\)
Height = Distance \(\times \tan \theta\)
Distance = \(\frac{\text{Height}}{\tan \theta}\)
Tip: In most height and distance problems, tangent is used because height and horizontal distance are involved.
Trigonometry Coverage Map
This section replaces the illustrative image and gives a quick guide for selecting the correct trigonometric tool in exam problems.
| Question Type | What to Use | Typical Clue |
|---|---|---|
| Angle conversion | Degree-radian conversion | \(180^\circ=\pi\) radians |
| Right triangle side finding | \(\sin\), \(\cos\), \(\tan\) | Perpendicular, base, hypotenuse |
| Identity simplification | Basic identities | \(\sin^2\theta+\cos^2\theta=1\) |
| Compound angle | Sum and difference formulae | \(\sin(A+B)\), \(\cos(A-B)\) |
| Double or half angle | Multiple/sub-multiple formulae | \(\sin 2A\), \(\cos 2A\), \(\sin \frac{A}{2}\) |
| Heights and distances | Usually tangent | Height, shadow, tower, distance |
| Triangle properties | Sine rule, cosine rule, area formula | Any general triangle, not necessarily right-angled |
Trigonometric Values of Standard Angles
Values of \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) for standard angles are very important for exams. These values should be memorized clearly.
| Angle \(\theta\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) |
|---|---|---|---|---|---|
| \(\sin \theta\) | 0 | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | 1 |
| \(\cos \theta\) | 1 | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | 0 |
| \(\tan \theta\) | 0 | \(\frac{1}{\sqrt{3}}\) | 1 | \(\sqrt{3}\) | Not defined |
| \(\cot \theta\) | Not defined | \(\sqrt{3}\) | 1 | \(\frac{1}{\sqrt{3}}\) | 0 |
| \(\sec \theta\) | 1 | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{2}\) | 2 | Not defined |
| \(\csc \theta\) | Not defined | 2 | \(\sqrt{2}\) | \(\frac{2}{\sqrt{3}}\) | 1 |
Important Trigonometric Functions
Apart from sine, cosine, and tangent, three reciprocal functions are also important: cosecant, secant, and cotangent.
| Function | Symbol | Related Function | Useful Relation |
|---|---|---|---|
| Sine | \(\sin \theta\) | Cosecant | \(\sin \theta \times \csc \theta = 1\) |
| Cosine | \(\cos \theta\) | Secant | \(\cos \theta \times \sec \theta = 1\) |
| Tangent | \(\tan \theta\) | Cotangent | \(\tan \theta \times \cot \theta = 1\) |
| Cosecant | \(\csc \theta\) | Sine | \(\csc \theta = \frac{1}{\sin \theta}\) |
| Secant | \(\sec \theta\) | Cosine | \(\sec \theta = \frac{1}{\cos \theta}\) |
| Cotangent | \(\cot \theta\) | Tangent | \(\cot \theta = \frac{1}{\tan \theta}\) |
Simple Trigonometric Identities
Trigonometric identities are equations that are true for all allowed values of the angle. They are useful for simplification and proof-based questions.
Pythagorean Identities
- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(1+\tan^2 \theta = \sec^2 \theta\)
- \(1+\cot^2 \theta = \csc^2 \theta\)
Ratio and Reciprocal Identities
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
Complementary Angle Identities
- \(\sin(90^\circ-\theta)=\cos \theta\)
- \(\cos(90^\circ-\theta)=\sin \theta\)
- \(\tan(90^\circ-\theta)=\cot \theta\)
- \(\sec(90^\circ-\theta)=\csc \theta\)
- \(\csc(90^\circ-\theta)=\sec \theta\)
- \(\cot(90^\circ-\theta)=\tan \theta\)
Useful Product Relations
- \(\sin \theta \times \csc \theta = 1\)
- \(\cos \theta \times \sec \theta = 1\)
- \(\tan \theta \times \cot \theta = 1\)
- \(\frac{\sin \theta}{\cos \theta} = \tan \theta\)
- \(\frac{\cos \theta}{\sin \theta} = \cot \theta\)
Sum and Difference Formulae
Sum and difference formulae are used to find trigonometric values of compound angles such as \(A+B\) and \(A-B\).
| Function | Sum Formula | Difference Formula |
|---|---|---|
| Sine | \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) | \(\sin(A-B)=\sin A\cos B-\cos A\sin B\) |
| Cosine | \(\cos(A+B)=\cos A\cos B-\sin A\sin B\) | \(\cos(A-B)=\cos A\cos B+\sin A\sin B\) |
| Tangent | \(\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\) | \(\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}\) |
Multiple and Sub-Multiple Angle Formulae
Multiple angle formulae are used for angles like \(2A\), while sub-multiple angle formulae are used for angles like \(\frac{A}{2}\).
| Formula Type | Formula | Use |
|---|---|---|
| Double Angle: Sine | \(\sin 2A=2\sin A\cos A\) | Used when product of sine and cosine appears. |
| Double Angle: Cosine | \(\cos 2A=\cos^2 A-\sin^2 A\) | Also written as \(1-2\sin^2 A\) or \(2\cos^2 A-1\). |
| Double Angle: Tangent | \(\tan 2A=\frac{2\tan A}{1-\tan^2 A}\) | Used in tangent double-angle problems. |
| Half Angle: Sine | \(\sin^2 \frac{A}{2}=\frac{1-\cos A}{2}\) | Used to find half-angle sine values. |
| Half Angle: Cosine | \(\cos^2 \frac{A}{2}=\frac{1+\cos A}{2}\) | Used to find half-angle cosine values. |
| Half Angle: Tangent | \(\tan \frac{A}{2}=\frac{\sin A}{1+\cos A}\) | Also \(\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}\). |
Inverse Trigonometric Functions
Inverse trigonometric functions are used to find an angle when the value of a trigonometric ratio is known. They are written as \(\sin^{-1}x\), \(\cos^{-1}x\), \(\tan^{-1}x\), etc.
| Inverse Function | Meaning | Example |
|---|---|---|
| \(\sin^{-1}x\) | Angle whose sine is \(x\). | If \(\sin \theta=\frac{1}{2}\), then \(\theta=\sin^{-1}\left(\frac{1}{2}\right)=30^\circ\). |
| \(\cos^{-1}x\) | Angle whose cosine is \(x\). | If \(\cos \theta=\frac{1}{2}\), then \(\theta=60^\circ\). |
| \(\tan^{-1}x\) | Angle whose tangent is \(x\). | If \(\tan \theta=1\), then \(\theta=45^\circ\). |
Properties of Triangles
Trigonometry is also used in general triangles, not only right-angled triangles. The sine rule, cosine rule, and area formula are useful for solving triangle problems.
| Property | Formula | Use |
|---|---|---|
| Angle Sum Property | \(A+B+C=180^\circ\) | Used in every triangle. |
| Sine Rule | \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) | Used when sides and opposite angles are involved. |
| Cosine Rule | \(a^2=b^2+c^2-2bc\cos A\) | Used when two sides and included angle are known. |
| Area of Triangle | \(\Delta=\frac{1}{2}bc\sin A\) | Used when two sides and included angle are known. |
| Right Triangle Property | \(H^2=P^2+B^2\) | Pythagoras theorem for right-angled triangles. |
Use of Trigonometric Tables
Trigonometric tables are used to find approximate values of trigonometric ratios for angles that are not standard angles such as \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\).
| Step | Action | Example Focus |
|---|---|---|
| Step 1 | Identify the trigonometric ratio required. | \(\sin\), \(\cos\), \(\tan\), etc. |
| Step 2 | Identify the given angle. | Example: \(37^\circ\), \(52^\circ\), \(75^\circ\). |
| Step 3 | Locate the degree row in the table. | Find the row for the given degree. |
| Step 4 | Use minutes column and mean difference if required. | Useful for angles like \(37^\circ 20'\). |
| Step 5 | Write the approximate value carefully. | Round off as required in the question. |
Heights and Distances
Heights and distances problems apply trigonometry to real-life situations. These questions usually involve towers, buildings, trees, poles, hills, shadows, observers, and angles of elevation or depression.
Angle of Elevation
When an observer looks upward at an object, the angle made by the line of sight with the horizontal line is called the angle of elevation.
Angle of Depression
When an observer looks downward at an object, the angle made by the line of sight with the horizontal line is called the angle of depression.
| Situation | Common Formula | Use |
|---|---|---|
| Height and horizontal distance are involved. | \(\tan \theta = \frac{\text{Height}}{\text{Distance}}\) | Most common height-distance problems. |
| Hypotenuse and height are involved. | \(\sin \theta = \frac{\text{Height}}{\text{Hypotenuse}}\) | Line of sight or slant distance problems. |
| Hypotenuse and horizontal distance are involved. | \(\cos \theta = \frac{\text{Distance}}{\text{Hypotenuse}}\) | Slant distance and ground distance problems. |
Step-by-Step Solving Method
| Step | Action | Example Focus |
|---|---|---|
| Step 1 | Check whether the problem is about value, identity, angle, triangle, or application. | Standard values, identities, heights and distances. |
| Step 2 | Check angle unit. | Degrees or radians. |
| Step 3 | For triangle questions, identify sides and angle. | Perpendicular, base, hypotenuse, opposite side. |
| Step 4 | Select the correct formula. | Basic ratio, identity, sum formula, double-angle formula, inverse function. |
| Step 5 | Substitute standard values and simplify. | Use \(30^\circ\), \(45^\circ\), \(60^\circ\) values carefully. |
Solved Examples
| Question | Method | Answer |
|---|---|---|
| Convert \(60^\circ\) into radians. | \[ 60^\circ=60\times \frac{\pi}{180}=\frac{\pi}{3} \] | \(\frac{\pi}{3}\) |
| Find \(\sin 30^\circ\), \(\cos 60^\circ\), and \(\tan 45^\circ\). |
From standard values: \(\sin 30^\circ=\frac{1}{2}\), \(\cos 60^\circ=\frac{1}{2}\), \(\tan 45^\circ=1\). |
\(\frac{1}{2},\frac{1}{2},1\) |
| If \(\sin \theta = \frac{3}{5}\), find \(\csc \theta\). | \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{3/5} = \frac{5}{3} \] | \(\frac{5}{3}\) |
| Simplify \(1-\sin^2 \theta\). | Using identity: \[ \sin^2 \theta+\cos^2 \theta=1 \] Therefore: \[ 1-\sin^2 \theta=\cos^2 \theta \] | \(\cos^2 \theta\) |
| Find \(\sin(45^\circ+30^\circ)\). | \[ \sin(A+B)=\sin A\cos B+\cos A\sin B \] \[ \sin 75^\circ= \frac{1}{\sqrt{2}}\cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}}\cdot \frac{1}{2} \] | \(\frac{\sqrt{3}+1}{2\sqrt{2}}\) |
| Find \(\cos 2A\) if \(\cos A=\frac{3}{5}\) and \(A\) is acute. | Since \(A\) is acute: \[ \sin A=\frac{4}{5} \] \[ \cos 2A=\cos^2 A-\sin^2 A = \frac{9}{25}-\frac{16}{25} = -\frac{7}{25} \] | \(-\frac{7}{25}\) |
| If \(\tan \theta=1\), find \(\theta\) for \(0^\circ<\theta<90^\circ\). | \[ \theta=\tan^{-1}(1) \] Since \(\tan 45^\circ=1\), \(\theta=45^\circ\). | \(45^\circ\) |
| A pole casts a shadow of 10 m when the angle of elevation of the sun is \(45^\circ\). Find the height of the pole. | \[ \tan 45^\circ=\frac{\text{Height}}{\text{Shadow}} \] \[ 1=\frac{h}{10} \] \[ h=10 \] | 10 m |
| A tower is observed from a point 20 m away. If the angle of elevation is \(60^\circ\), find the height of the tower. | \[ \tan 60^\circ=\frac{h}{20} \] \[ \sqrt{3}=\frac{h}{20} \] \[ h=20\sqrt{3} \] | \(20\sqrt{3}\) m |
| In triangle \(ABC\), if \(A=60^\circ\), \(b=8\), \(c=10\), find area. | \[ \Delta=\frac{1}{2}bc\sin A \] \[ \Delta=\frac{1}{2}\times 8\times 10\times \frac{\sqrt{3}}{2} = 20\sqrt{3} \] | \(20\sqrt{3}\) square units |
Note: Always check whether the question asks for a value, an identity, a side of a triangle, a height, a distance, an inverse angle, or a reciprocal ratio.
Common Traps and Shortcuts
Common Traps
- Confusing degrees and radians.
- Confusing \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).
- Using wrong standard angle values.
- Forgetting that \(\tan 90^\circ\) is not defined.
- Confusing inverse sine \(\sin^{-1}x\) with reciprocal sine.
- Confusing angle of elevation with angle of depression.
- Not drawing a diagram in height and distance problems.
- Using hypotenuse as base or perpendicular incorrectly.
- Applying identities without checking the expression carefully.
Useful Shortcuts
- For heights and distances, first try \(\tan \theta\).
- Remember \(\sin\) values increase from \(0^\circ\) to \(90^\circ\).
- Remember \(\cos\) values decrease from \(0^\circ\) to \(90^\circ\).
- \(\tan 45^\circ = 1\) is useful in quick problems.
- Convert all ratios into sine and cosine for simplification.
- Use complementary identities for \(90^\circ-\theta\) questions.
- Use \(\sin(A+B)\) when a value like \(\sin 75^\circ\) appears.
- Use \(180^\circ=\pi\) radians for angle conversion.
Practice
A) Multiple Choice Questions
-
What is the value of \(\sin 30^\circ\)?
0 \(\frac{1}{2}\) \(\frac{1}{\sqrt{2}}\) 1
-
What is the radian measure of \(90^\circ\)?
\(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{2}\) \(\pi\)
-
Which identity is correct?
\(\sin^2 \theta - \cos^2 \theta = 1\) \(\sin^2 \theta + \cos^2 \theta = 1\) \(\tan^2 \theta + \sec^2 \theta = 1\) \(\cot^2 \theta - 1 = \csc^2 \theta\)
-
What is the reciprocal of \(\cos \theta\)?
\(\sin \theta\) \(\tan \theta\) \(\sec \theta\) \(\cot \theta\)
-
If the height of a tower is 30 m and the distance from its base is 30 m, then the angle of elevation is:
30° 45° 60° 90°
B) Solve the Higher-Order Problems
- Convert \(45^\circ\) into radians. (Hint: Multiply by \(\frac{\pi}{180}\).)
- If \(\sin \theta = \frac{4}{5}\), find \(\csc \theta\). (Hint: \(\csc \theta = \frac{1}{\sin \theta}\).)
- Simplify \(1-\cos^2 \theta\). (Hint: Use \(\sin^2 \theta + \cos^2 \theta = 1\).)
- A tree casts a shadow of 15 m when the angle of elevation of the sun is \(45^\circ\). Find the height of the tree. (Hint: Use \(\tan 45^\circ = \frac{\text{Height}}{\text{Shadow}}\).)
- Find the value of \(\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ\). (Hint: Substitute standard values or use \(\sin(A+B)\).)
C) Match the Concept with the Correct Rule
| Concept | Correct Rule / Meaning |
|---|---|
| Sine | Perpendicular divided by hypotenuse |
| Cosine | Base divided by hypotenuse |
| Tangent | Perpendicular divided by base |
| Secant | Reciprocal of cosine |
| Cosecant | Reciprocal of sine |
| Angle of Elevation | Angle made while looking upward from the horizontal line |
| Sine Rule | \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) |
| Inverse Trigonometric Function | Gives the angle when the trigonometric ratio is known |
Trigonometry Reminder
Trigonometry is mainly about understanding the relationship between sides and angles. Students should master degree-radian conversion, standard angle values, basic ratios, reciprocal ratios, simple identities, sum and difference formulae, double-angle and half-angle formulae, inverse trigonometric functions, properties of triangles, trigonometric tables, and simple heights and distances applications.
Task: Create five trigonometry questions using one question each from angle conversion, standard values, identities, inverse trigonometric functions, triangle properties, and heights and distances.
Show Suggested Answers
Multiple Choice
-
\(\frac{1}{2}\)
From standard values, \(\sin 30^\circ = \frac{1}{2}\). -
\(\frac{\pi}{2}\)
\(90^\circ=90\times\frac{\pi}{180}=\frac{\pi}{2}\). -
\(\sin^2 \theta + \cos^2 \theta = 1\)
This is the basic Pythagorean trigonometric identity. -
\(\sec \theta\)
Reciprocal of \(\cos \theta\) is \(\sec \theta\). -
45°
\(\tan \theta = \frac{30}{30}=1\). Since \(\tan 45^\circ=1\), the angle is \(45^\circ\).
Higher-Order Problems
- \[ 45^\circ=45\times\frac{\pi}{180}=\frac{\pi}{4} \] Answer = \(\frac{\pi}{4}\).
- \[ \csc \theta=\frac{1}{\sin \theta}=\frac{1}{4/5}=\frac{5}{4} \] Answer = \(\frac{5}{4}\).
-
Using \(\sin^2 \theta+\cos^2 \theta=1\),
\(1-\cos^2 \theta=\sin^2 \theta\).
Answer = \(\sin^2 \theta\). -
Shadow = 15 m and angle = \(45^\circ\).
\[ \tan 45^\circ=\frac{h}{15} \] \[ 1=\frac{h}{15} \] Therefore, \(h=15\).
Answer = 15 m. - \[ \sin 60^\circ \cos 30^\circ+\cos 60^\circ \sin 30^\circ \] \[ =\frac{\sqrt{3}}{2}\times\frac{\sqrt{3}}{2} + \frac{1}{2}\times\frac{1}{2} = \frac{3}{4}+\frac{1}{4}=1 \] Answer = 1.
Concept Matching
- Sine → Perpendicular divided by hypotenuse
- Cosine → Base divided by hypotenuse
- Tangent → Perpendicular divided by base
- Secant → Reciprocal of cosine
- Cosecant → Reciprocal of sine
- Angle of Elevation → Angle made while looking upward from the horizontal line
- Sine Rule → \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
- Inverse Trigonometric Function → Gives the angle when the trigonometric ratio is known
Clue Explanation
Trigonometry questions are solved by first identifying the angle unit, triangle type, known quantities and required result. In identities, convert expressions into sine and cosine where possible. In height and distance problems, draw a diagram before applying the formula.
Exam tips
- Use \(180^\circ=\pi\) radians for conversion.
- Memorize standard values from \(0^\circ\) to \(90^\circ\).
- Remember that \(\tan 90^\circ\) is not defined.
- Use \(\csc\) for cosecant in formulas.
- Use \(\tan \theta\) for most height-distance problems.
- Draw diagrams for angle of elevation and depression.
- Use identities to simplify expressions.
- Use sine rule and cosine rule for general triangles.
- For \(0^\circ < \theta < 90^\circ\), all trigonometric ratios are positive.