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ICSE Class 9 Math Revision Notes Chapter 23 - Trigonometrical Ratios of Standard Angles

Last updated date: 02nd Aug 2024
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Revision Notes for ICSE Class 9 Mathematics Chapter 23 - Free PDF Download

Trigonometry is a study of the connection of angles, heights and lengths. It guides in locating the angles and missing sides of a triangle with the help of trigonometric ratios. These angles are measured in either radians or degrees.

Competitive Exams after 12th Science

The six trigonometric ratios are essentially conveyed in terms of the right-angled triangle

The six trigonometric ratios are essentially conveyed in terms of the right-angled triangle

Trignometric Ratios Table (Standard Angles)

 Angle = $\angle c$ 0o 30o 45o 60o 90° Sin C 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 Cos C 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 Tan C 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ Not Defined Cosec C Not Defined 2 $\sqrt{2}$ $\frac{2}{\sqrt{3}}$ 1 Sec C 1 $\frac{2}{\sqrt{3}}$ $\sqrt{2}$ 2 Not Defined Cot C Not Defined $\sqrt{3}$ 1 $\frac{1}{\sqrt{3}}$ 0

• Write the integers 0, 1, 2, 3 and 4

• Divide these integers by 4

• Find the square root of the rational numbers obtained in the above step

• The values in the above step are the values of sin 0o, sin 30o, sin 45o, sin 60o and sin 90o

• Writing them in reverse gives the corresponding value of cos

• Dividing the sin values by their corresponding cos values gives the corresponding values of tan

The values of cosec, sec and cot are the reciprocals of sin, cos and tan

7 Easy And Simple Steps To Learn Trigonometry

Step 1. Study all the basics of trigonometric angles.

Step 2. Study right-angle triangle concepts.

It is a triangle with one angle of 90 degrees. The longest side is called the hypotenuse. The other two sides are the adjacent side and the opposite side. The following are the three basic functions in trigonometry.

sine = (opposite side) / hypotenuse

cosine = (adjacent side) / hypotenuse

tangent = (opposite side) / (adjacent side)

Step 3. Understand the Pythagoras theorem.

Hypotenuse2 = Base2 + Altitude2

Step 4.  Learn the Sine rule and Cosine rule.

Sine rule: a/ sin A = b/ sin B = c/ sin C

Cosine rule: c2 = a2 + b2 – 2ab cos C

Step 5. List all the important identities of trigonometry.

Fundamental identities:

sin2θ + cos2θ = 1

1 + tan2θ = sec2θ

1 + cot2θ = cosec2 θ

Reciprocal formulas:

1/sin x = cosec x

1/cos x = sec x

1/tan x = cot x

Step 6. Remember the trigonometry table.

The table contains the values of trigonometric ratios (sine, cosine, tangent, cotangent, secant, and cosecant) of standard angles.

Step 7. Be thorough with the trigonometric formulas. Even if you forget the table, you can easily calculate the values using the formulas.

sin x = cos (90°-x)

cos x = sin (90°-x)

tan x = cot (90°-x)

cot x = tan (90°-x)

sec x = cosec (90°-x)

cosec x = sec (90°-x)

Trigonometry appears to be easy only when the students are thorough with all the rules and formulas. Here are some questions and it’s solutions to help them to improve their problem-solving speed.

Problem-solving:

Find the value of:

(i) sin 30o cos 30o

$Sin 30^{\circ}Cos 30^{\circ}= \frac{1}{2}.\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}$

(ii) tan 30o tan 60o

$tan 30^{\circ}tan 60^{\circ}= \frac{1}{\sqrt{3}}(\sqrt{3})=1$

(iii) cos2 60o + sin2 30o

$Cos^{2} 60^{\circ}+Sin^{2} 30^{\circ}=\left ( \frac{1}{2} \right )^{2}+\left ( \frac{1}{2} \right )^{2}=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$

(iv) cosec2 60o - tan2 30o

$Cosec^{2} 60^{\circ}-tan^{2} 30^{\circ}=\left ( \frac{2}{\sqrt{3}} \right )^{2}+\left ( \frac{1}{\sqrt{3}} \right )^{2}=\frac{4}{3}-\frac{1}{3}=1$

(v) sin2 30o + cos2 30o + cot2 45o

$Sin^{2} 30^{\circ}+Cos^{2} 30^{\circ}+Cot^{2}45^{\circ}=\left ( \frac{1}{2} \right )^{2}+\left ( \frac{\sqrt{3}}{2} \right )^{2}+1^{2}=\frac{1}{4}+\frac{3}{4}+1=2$

(vi) cos2 60o + sec2 30o + tan2 45o.

$Cos^{2} 60^{\circ}+Sec^{2} 30^{\circ}+tan^{2}45^{\circ}=\left ( \frac{1}{2} \right )^{2}+\left ( \frac{2}{\sqrt{3}} \right )^{2}+1^{2}$

$\frac{1}{4}+\frac{4}{3}+1$

$\frac{3+16+12}{2}$

$\frac{31}{12}$

$2\frac{7}{12}$

(i) If sin x = cos x and x is acute, state the value of x.

The angle, x is acute and hence we have, 0 < x

We Know That

$Cos^{2}X +Sin^{2}x=1$

$2Sin^{2}x=1$                 [since cosx = sin x]

Sin x=$\frac{1}{\sqrt{2}}$

x= 45

(ii) If sec A = cosec A and 0o A 90o, state the value of A.

Sec A = Cosec A

Cos A = Sin A

$Cos^{2}$ A = 1 - Cos2 A

2Cos2A = 1

A= 45o

(iii) If tan = cot and 0o 90o, state the value of.

Tan θ = Cot θ

Tan θ = 1/tan θ

Tan2 θ = 1

Tan θ = 1

Tan θ =  tan 45o

θ = 45o

(iv) If sin x = cos y; write the relation between x and y, if both the angles x and y are acute.

Sin x = cos y = sin (90o - y)

If x and y are acute angles,

X = 90o - y

X + y = 90o

Trigonometry is used to set directions such as the north-south and east-west. It shows what direction to use with the compass that leads in a straight direction. It is used for navigation to locate a place. It is also applied to find the distance of the shore from a point in the sea.

FAQs on ICSE Class 9 Math Revision Notes Chapter 23 - Trigonometrical Ratios of Standard Angles

1. What is trigonometry, according to chapter 23 of Class 9 ICSE Maths?

Trigonometry is a section of mathematics in geography, which handles the sides and angles of a right-angled triangle. Trigonometric ratios are assessed with the sides and angles. For any trigonometry-related queries of Class 9 ICSE maths, head over to the website of Vedantu. Here, you can find all the reading materials that will enhance your understanding of the chapter and further, help you to make the basics of the chapter strong.

2. What are the trigonometric ratios according to chapter 23 of Class 9 ICSE Maths?

The names and abbreviations of trigonometric ratios are sine (sin), secant (sec), cosine (cos), cosecant (cosec), cotangent (cot) and tangent (tan). The standard angles for these trigonometric ratios are 0°, 30°, 45°, 60° and 90°. These angles can also be depicted as radians which are 0, π/2, π/3,  π/4, π/6.

3. According to chapter 23 of Class 9 ICSE Maths, where are trigonometric ratios used?

The trigonometric ratios can be applied to find numerous pieces of information which are concerned with specific functions of angles and their application to calculations, and one of their main motives is to help figure out triangles. A triangle can be solved by finding the measures of all the angles and the lengths of the sides.

4. When can we use sine law, according to chapter 23 of Class 9 ICSE Maths?

The sine rule is applied when we have either a) two angles and one side, or b) two sides and a non-included angle. The cosine rule is applied when having either a) three sides or b) two sides and the included angle. Download the Vedantu app for any queries that you have regarding the chapter. You can sign up for free classes wih our experts as well for a proper grasp of the concepts.

5. According to chapter 23 of Class 9 ICSE Maths, how do you find a hypotenuse?

The hypotenuse is known as the longest side of a right-angled triangle. To figure out the longest side we apply the hypotenuse formula that can be easily driven from the Pythagoras theorem, (Hypotenuse)2 = (Base)2 + (Altitude)2. Hypotenuse formula = √((base)2 + (height)2) (or) c = √(a2 + b2).