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

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

Trignometric Ratios Table (Standard Angles)

An Easy Way To Learn These Trigonometric Ratios 

• 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 30^o cos 30^o

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

(ii) tan 30^o tan 60^o

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

(iii) cos² 60^o + sin² 30^o

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

(iv) cosec² 60^o - tan² 30^o

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

(v) sin² 30^o + cos² 30^o + cot² 45^o

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

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

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

\[\frac{1}{4}+\frac{4}{3}+1\]

\[\frac{3+16+12}{2}\]

\[\frac{31}{12}\]

\[2\frac{7}{12}\]

(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= 45^o

(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 45^o

θ = 45^o

(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 (90^o - y)

If x and y are acute angles,

X = 90^o - y

X + y = 90^o

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.