Trigonometric Ratios of Standard Angles: Values & Derivation

Trigonometric Ratios of Standard Angles are the six trigonometric ratios for the standard angles 0°, 30°, 45°, 60° and 90°. Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of a right-angled triangle.

• Trigonometric Ratios of Standard Angles are used to relate the sides of a right triangle to the angles of the triangle. 
• There are six trigonometric ratios namely Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant.
• Sine, Cosine, and Tangent are primary trigonometric ratios.
• Cosec, Sec, and Cot are other important trigonometric ratios that can be derived using sin, cos, and tan respectively.
• These ratios are defined for acute angle (Less than 90^o) in a right-angled triangle.
• They are typically denoted using the symbols sin θ, cos θ, and tan θ, where θ represents the angle measure. 
• The standard angles of trigonometric ratios are 0°, 30°, 45°, 60° and 90°.
• Standard angles can also be expressed in Radians such as 0, π/6, π/4, π/3, and π/2. 

Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions

Key Terms: Trigonometric Ratios, Right-angled Triangle, Hypotenuse, Sine, Cosine, Tangent, Cotangent, Secant, Cosecant

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What are Trigonometric Ratios?

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Trigonometric Ratios are the ratios of the length of sides of a right-angled triangle in Trigonometry. Trigonometric ratios relate the ratio of sides of a right triangle to the respective angle.

There are six Trigonometric Ratios namely: 

1. Sine (sin)
2. Cosine (cos)
3. Tangent (tan)
4. Cotangent (cot)
5. Secant (sec)
6. Cosecant (cosec)

Sine (sin), Cosine (cos), and Tangent (tan) are the primary trigonometric ratios while Cotangent (cot), Secant (sec), and Cosecant (cosec) can be derived from sin, cos, and tan respectively.

Trigonometric Functions Detailed Video Explanation

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Trigonometric Ratios Definition

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In trigonometry, there are six trigonometric ratios namely Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. They are used to determine the ratios of any two sides of a right-angled triangle in terms of their respective angles. The values of trigonometric ratios can be calculated using the measure of an acute angle, θ.

Consider a right-angled triangle ABC with an acute angle, θ at C.

Right-angled Triangle ABC

The six trigonometric ratios can be thus defined as: 

• Sine: Sine Ratio is defined as the ratio of the perpendicular to the hypotenuse. In the given triangle, sin θ = AB/AC.
• Cosine: Cosine Ratio is defined as the ratio of the base to the hypotenuse. In the given triangle, cos θ = BC/AC.
• Tangent: Tangent Ratio is defined as the ratio of the perpendicular to the base. In the given triangle, tan θ = AB/BC.
• Cosecant: Cosecant Ratio is defined as the ratio of the hypotenuse to the perpendicular. In the given triangle, cosec θ = AC/AB.
• Secant: Secant Ratio is defined as the ratio of the hypotenuse to the base. In the given triangle, sec θ = AC/BC.
• Cotangent: Cotangent Ratio is defined as the ratio of the base to the perpendicular. In the given triangle, cot θ = BC/AB.

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Trigonometric Ratios Formulas

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Trigonometric Ratios are defined as the ratio of any two sides of the right-angled triangle. Given below are the basic trigonometric ratios formulas: 

• sin θ = Perpendicular/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Perpendicular/Base
• sec θ = Hypotenuse/Base
• cosec θ = Hypotenuse/Perpendicular
• cot θ = Base/Perpendicular

It can be noticed that some trigonometric ratios are reciprocal of others such as sin θ is a reciprocal of cosec θ, cos θ is a reciprocal of sec θ, tan θ is a reciprocal of cot θ, and vice-versa. Thus, it results in another set of trigonometric ratios formulas:

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Read More: Trigonometric Formulas

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Trigonometric Ratios of Standard Angles

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Sine, Cosine, Tangent, Cotangent, Secant and Cosecant are the six trigonometric ratios. The standard angles for these trigonometric ratios include 0°, 30°, 45°, 60° and 90°. The values of the standard angles of trigonometric ratios can easily be checked from the Trigonometry Table. 

Here are the values of Trigonometric Ratios of Standard Angles in Trigonometry: 

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Trigonometric Ratios of Standard Angles Derivation

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Trigonometric Ratios of Standard Angles can be derived using sine, cosine, and tangent ratios.

• Sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. 
• Cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. 
• Tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. 

Using these definitions, it is possible to derive the trigonometric ratios for standard angles by constructing a right triangle with specific angle measures and calculating the corresponding side lengths. 

Trigonometrical Ratio of 45°

Let ABC be a right-angled isosceles triangle with ∠B = 90° and ∠A = ∠C = 45°. Let BC = AC = α.

Right-angled Isosceles Triangle ABC

Thus,

H² = P² + B²

(AC)² = (AB)² + (BC)²

(AC)² = α² + α²

(AC)² = 2 α²

∴ AC = α√2

Using ∠C as a point of reference, the trigonometric ratios for 45° are: 

• Sin 45 = AB/AC = α/α√2 = 1/√2
• Cot 45 = BC/AC = α/α√2 = 1/√2
• Tan 45 = AB/BC = α/α = 1
• Cosec 45 = AC/AB = α√2/α = √2
• Sec 45 = AC/AB = α√2/α = √2
• Cot 45 = BC/AB = α/α = 1

Trigonometrical Ratio of 60°

Assume ABC is an equilateral triangle with ∠A = ∠B = ∠C = 60°

Equilateral Triangle ABC

Thus, AB = BC = CA = 2α

Let's now draw AD perpendicular to BC so that it looks like this:

Equilateral Triangle ABC

Therefore, BD = DC = a , And ∠BAD = ∠DAC = 30°

Now, in right-angled ΔADC, 

(AC)² = (AD)² + (DC)²

(AD)² = (AC)² + (DC)²

= (2α)² - (α)² = 4α² - α² = 3 α²

∴ AD = α√3

Now, to find the trigonometric ratio for 60°, let's take ∠C = 60° as the reference angle, we get,

• Sin 60 = AD/AC = α√3/2α = √3/2
• Cos 60 = DC/AC = α/2α = ½
• Tan 60 = AD/DC = α√3/2α = √3
• Cosec 60 = AC/AD = 2α/α√3 = 2/√3
• Sec 60 = AC/DC = 2α/α = 2
• Cot 60 = DC/AD = α/α√3 = 1/√3

Read More: ​Trigonometric Functions Important Questions

Trigonometrical Ratio of 30°

Using ∠DAC = 30° in the above triangle as the reference angle, the trigonometric ratio for 30° can be obtained.

• Sin 30 = DC/AC = α/2α = 1/2
• Cos 30 = = AD/AC = α√3/2α = √3/2
• Tan 30 = = DC/AD = α/α√3 = 1/√3
• Cosec 30 = AC/DC = 2α/α = 2
• Sec 30 = AC/AD = 2α/α√3 = 2/√3
• Cot 30 = AD/DC = α√3/2α = √3

Trigonometrical Ratio of 0°

Let ABC be a right-angled triangle with ∠B = 90° and ∠C = θ.

Right-angled Triangle ABC

If θ tends to 0°

i.e., AC corresponds with BC at θ → 0°.

or, AC ≈ BC

so, AC = BC = a (say)

Using Pythagoras' theorem now

h² = p² + b²

or, (AC)² = (AB)² + (BC)²

or, a² = (AB)² + a²

or, (AB)² = a² - a² = 0

∴ AB = 0

Thus, the trigonometric ratios for 0° are: 

• Sin 0 = p/h = AB/AC = 0/α = 0
• Cos 0 = b/h = BC/AC = α/α =1
• Tan 0 = p/b = AB/BC = 0/α = 0
• Cosec 0 = h/p = AC/AB = a/0 = ∞
• Sec 0 = h/b = AC/BC = α/α = 1
• Cot 0 = b/p = BC/AB = α/0 = ∞

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Trigonometrical Ratio of 90° 

Let ABC be a right-angled triangle with the reference angles ∠B = 90° and ∠C = θ.

Right-angled Triangle ABC

If θ tends to 90°

i.e., AC corresponds with BC at θ → 90°.

or, AC ≈ BC

so, AC= BC = a (say)

Using Pythagoras' theorem now

H² = P² +B²

(AC)² = (AB)² + (BC)²

a² = (BC)² + a²

(BC)² = a² - a² = 0

∴ BC = 0

Thus, the trigonometric ratios for 90° are: 

• Sin 90 = p/h = AB/AC = α/α = 1
• Cos 0 = b/h = BC/AC = 0/α = 0
• Tan 0 = p/b = AB/BC = α/0 = ∞
• Cosec 0 = h/p = AC/AB = α/α = 1
• Sec 0 = h/b = AC/BC = α/0 = ∞
• Cot 0 = b/p = BC/AB = 0/α = 0

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Things to Remember 

• Trigonometric Ratios are ratios used to relate the ratio of sides of a right triangle to the respective angle. 
• Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant are six major trigonometric ratios. 
• Trigonometric Ratios of Standard Angles are defined for acute angles 0°, 30°, 45°, 60° and 90°. 
• Sine (sin θ) is the ratio of the perpendicular to the hypotenuse.
• Cosine (cos θ) is the ratio of the adjacent side to the hypotenuse. 
• Tangent (tan θ) is the ratio of the perpendicular to the adjacent side. 
• Cosecant (cosec θ) is the ratio of the hypotenuse to the perpendicular.
• Secant (sec θ) is the ratio of the hypotenuse to the adjacent side. 
• Cotangent (cot θ) is the ratio of the adjacent side to the perpendicular.

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