Maharashtra Board Class 9 Maths Part II Chapter 8 Trigonometry PDF Download

Trigonometry

Introduction To Trigonometry

We can measure distances by using a rope or by walking on ground. But how do we measure the distance between a ship and a light house? How do we measure the height of a tall tree?

Look at the pictures above. The questions in the pictures are about mathematics. Trigonometry is a branch of mathematics. It is useful to find answers to such questions. Trigonometry is used in different branches of Engineering, Astronomy, and Navigation.

The word Trigonometry comes from three Greek words. 'Tri' means three. 'Gona' means sides. 'Metron' means measurements.

We have studied triangles. The subject trigonometry starts with right angled triangles. It also uses the Pythagoras theorem and similar triangles. So we will recall these topics.

In triangle ABC, angle B is a right angle. The side AC is opposite to B. This side is the hypotenuse. The side opposite to angle A is BC. The side opposite to angle C is AB.

Using Pythagoras' theorem, we can write this statement for the triangle:

\[(AB)^2 + (BC)^2 = (AC)^2\]

Teacher's Note

Trigonometry helps us find heights and distances without actually measuring them. For example, builders use trigonometry to find the height of a building or bridge.

Exam Trick

Remember: Trigonometry = finding heights and distances using angles and sides of triangles. Think of it like measuring things from far away using math!

Points to Remember

Trigonometry is used to measure heights and distances.
The word comes from three Greek words meaning three, sides, and measurements.
It starts with right angled triangles and Pythagoras' theorem.
We use angles and sides to find unknown measurements.

If triangle ABC is similar to triangle PQR, then their matching sides are in the same ratio.

So \[\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\]

How To Find The Height Of A Tall Tree Using Similar Triangles

This experiment can be done on a clear sunny day.

Look at the figure given with this lesson.

The height of the tree is QR. The height of the stick is BC.

Push a stick into the ground as shown in the figure. Measure its height. Also measure the length of its shadow. Then measure the length of the shadow of the tree. The rays of sunlight are parallel. So triangle PQR and triangle ABC are similar triangles. The sides of similar triangles are in the same ratio.

So we get \[\frac{QR}{PR} = \frac{BC}{AC}\]

Therefore, we get this equation:

\[\text{height of the tree} = QR = \frac{BC}{AC} \times PR\]

We know the values of PR, BC, and AC. When we put these values in this equation, we get the length of QR. This is the height of the tree.

Teacher's Note

You can actually try this activity in your school. On a sunny day, measure a stick and its shadow, then use the tree's shadow to find its height. It really works!

Exam Trick

Remember the formula: Height of tree = (Stick height / Stick shadow) × Tree shadow. Keep the formula simple and you will not forget it.

Points to Remember

Similar triangles have sides in the same ratio.
Sunlight rays are parallel, which makes the shadows useful.
We can find the height of a tree without climbing it.
We need to measure the stick, stick shadow, and tree shadow.
The formula is: Tree height = (BC/AC) × PR.

Activity: You can do this activity and find the height of a tall tree in your area. If there is no tree in your school, then find the height of a lamp post or pole.

Use Your Brain Power: It is better to do this experiment between 11:30 am and 1:30 pm. Can you tell why?

Terms Related To Right Angled Triangle

In right angled triangle ABC, angle B equals 90 degrees. Angles A and C are acute angles.

The opposite side of angle A is the side BC. The adjacent side of angle A is the side AB. The hypotenuse is the side AC.

The opposite side of angle C is the side AB. The adjacent side of angle C is the side BC. The hypotenuse is the side AC.

Example: In right angled triangle PQR, write the following:

The side opposite to angle P is........

The side opposite to angle R is........

The side adjacent to angle P is........

The side adjacent to angle R is........

Trigonometric Ratios

In the figure, some right angled triangles are shown. Angle B is their common angle. So all right angled triangles are similar.

Triangle PQB is similar to triangle ACB.

So \[\frac{PB}{AB} = \frac{PQ}{AC} = \frac{BQ}{BC}\]

This gives us \[\frac{PQ}{AC} = \frac{PB}{AB}\]

By rearranging, \[\frac{PQ}{PB} = \frac{AC}{AB}\]

This is called the sine ratio of angle B. We write it as sin B.

\[\sin B = \frac{\text{opposite side of } B}{\text{hypotenuse}} = \frac{PQ}{PB} = \frac{AC}{AB}\]

Also, \[\frac{BQ}{PB} = \frac{BC}{AB} = \frac{\text{adjacent side of } B}{\text{hypotenuse}}\]

This is called the cosine ratio of angle B. We write it as cos B.

\[\cos B = \frac{\text{adjacent side of } B}{\text{hypotenuse}} = \frac{BQ}{PB} = \frac{BC}{AB}\]

Also, \[\frac{PQ}{BQ} = \frac{AC}{BC} = \frac{\text{opposite side of } B}{\text{adjacent side of } B}\]

This is called the tangent ratio of angle B. We write it as tan B.

\[\tan B = \frac{\text{opposite side of } B}{\text{adjacent side of } B} = \frac{PQ}{BQ} = \frac{AC}{BC}\]

Sometimes we write the measures of acute angles of a right angled triangle using Greek letters. We use θ (Theta), α (Alpha), β (Beta) and so on.

In the triangle ABC, the measure of acute angle C is shown by the letter θ. So we can write the ratios sin C, cos C, tan C as sin θ, cos θ, tan θ.

Teacher's Note

Sin, cos, and tan are just short names for sine, cosine, and tangent. Think of them like abbreviations that make writing easier, just like SMS is short for Short Message Service.

Exam Trick

Remember SOH-CAH-TOA: Sin = Opposite over Hypotenuse, Cos = Adjacent over Hypotenuse, Tan = Opposite over Adjacent. This one phrase helps you remember all three ratios!

Points to Remember

Sin means the ratio of opposite side to hypotenuse.
Cos means the ratio of adjacent side to hypotenuse.
Tan means the ratio of opposite side to adjacent side.
All three are called trigonometric ratios.
They are used to find missing sides and angles in triangles.

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