Trigonometric Ratios, Degree and Radian Measures, Standard Identities

Welcome, students! Today, we embark on a crucial journey into the world of trigonometry. This branch of mathematics deals with the relationships between the angles and sides of triangles. Understanding trigonometric ratios, how to measure angles in degrees and radians, and the fundamental identities will be your bedrock for solving complex problems not just in mathematics, but also in physics and engineering. Let's break it down step by step.

Understanding Trigonometric Ratios

Trigonometric ratios are essentially ratios of the sides of a right-angled triangle with respect to its acute angles. Consider a right-angled triangle ABC, where angle B is 90 degrees. Let angle C be denoted by θ (theta).

In relation to angle θ:

• The side opposite to angle θ is BC.
• The side adjacent to angle θ is AB.
• The hypotenuse (the side opposite the right angle) is AC.

The six fundamental trigonometric ratios are defined as follows:

1. Sine (sin θ): The ratio of the length of the side opposite the angle to the length of the hypotenuse.

   sin θ = Opposite / Hypotenuse = BC / AC

2. Cosine (cos θ): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

   cos θ = Adjacent / Hypotenuse = AB / AC

3. Tangent (tan θ): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

   tan θ = Opposite / Adjacent = BC / AB

4. Cosecant (csc θ or cosec θ): The reciprocal of sine θ.

   csc θ = 1 / sin θ = Hypotenuse / Opposite = AC / BC

5. Secant (sec θ): The reciprocal of cosine θ.

   sec θ = 1 / cos θ = Hypotenuse / Adjacent = AC / AB

6. Cotangent (cot θ): The reciprocal of tangent θ.

   cot θ = 1 / tan θ = Adjacent / Opposite = AB / BC

We can also express tangent and cotangent in terms of sine and cosine:

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

Example:

In a right-angled triangle XYZ, with angle Y = 90 degrees, if XY = 5 cm and YZ = 12 cm, find the trigonometric ratios for angle Z.

First, we need to find the hypotenuse XZ using the Pythagorean theorem: XZ² = XY² + YZ².

XZ² = 5² + 12² = 25 + 144 = 169.

XZ = √169 = 13 cm.

Now, for angle Z:

• Opposite side = XY = 5 cm
• Adjacent side = YZ = 12 cm
• Hypotenuse = XZ = 13 cm

Therefore:

• sin Z = Opposite / Hypotenuse = 5 / 13
• cos Z = Adjacent / Hypotenuse = 12 / 13
• tan Z = Opposite / Adjacent = 5 / 12
• csc Z = 1 / sin Z = 13 / 5
• sec Z = 1 / cos Z = 13 / 12
• cot Z = 1 / tan Z = 12 / 5

Degree and Radian Measures of Angles

Angles can be measured in two primary units: degrees and radians. Understanding the relationship between them is vital.

Degree Measure:

A degree is defined as 1/360th of a full circle. A full circle is 360°. A right angle is 90°, and a straight angle is 180°.

We use the symbol '°' to denote degrees. For example, 30°, 45°, 60°, 90°.

Degrees can be further divided into minutes (') and seconds ('').

• 1 degree = 60 minutes (1° = 60')
• 1 minute = 60 seconds (1' = 60'')

Radian Measure:

A radian is a measure of angle defined in terms of the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Let 'r' be the radius of a circle and 's' be the length of an arc. The angle θ subtended by the arc at the center in radians is given by:

θ (in radians) = Arc Length / Radius = s / r

Relationship between Degrees and Radians:

A full circle is 360°.

The circumference of a circle is 2πr, where 'r' is the radius.

The angle subtended by the full circle (circumference) at the center in radians is:

Angle = Circumference / Radius = 2πr / r = 2π radians

Therefore, we have the fundamental relationship:

360° = 2π radians

Dividing by 2, we get:

180° = π radians

From this, we can derive conversion formulas:

• To convert degrees to radians: Multiply the degree measure by π / 180.
• To convert radians to degrees: Multiply the radian measure by 180 / π.

Example Conversions:

1. Convert 120° to radians:

   120° * (π / 180) = 120π / 180 = (2/3)π radians

2. Convert π/5 radians to degrees:

   (π / 5) * (180 / π) = 180 / 5 = 36°

Standard Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables for which the expressions in the equation are defined. They are fundamental tools for simplifying trigonometric expressions and solving equations.

1. Pythagorean Identities:

These are derived from the Pythagorean theorem (a² + b² = c²) applied to a right-angled triangle and the unit circle. The unit circle is a circle with radius 1 centered at the origin (0,0) in a Cartesian coordinate system. For any point (x, y) on the unit circle corresponding to an angle θ, we have x = cos θ and y = sin θ. Since x² + y² = 1 (radius squared), we get:

Identity 1: sin² θ + cos² θ = 1

This is the most fundamental identity. It holds true for any angle θ.

To derive the other two Pythagorean identities, we can divide the first identity by cos² θ and sin² θ, respectively (assuming cos θ ≠ 0 and sin θ ≠ 0).

Dividing sin² θ + cos² θ = 1 by cos² θ:

(sin² θ / cos² θ) + (cos² θ / cos² θ) = 1 / cos² θ

tan² θ + 1 = sec² θ

Identity 2: 1 + tan² θ = sec² θ

Dividing sin² θ + cos² θ = 1 by sin² θ:

(sin² θ / sin² θ) + (cos² θ / sin² θ) = 1 / sin² θ

1 + cot² θ = csc² θ

Identity 3: 1 + cot² θ = csc² θ

2. Reciprocal Identities:

As defined earlier, these relate the six trigonometric functions to each other.

• sin θ = 1 / csc θ => csc θ = 1 / sin θ
• cos θ = 1 / sec θ => sec θ = 1 / cos θ
• tan θ = 1 / cot θ => cot θ = 1 / tan θ

3. Quotient Identities:

These express tangent and cotangent in terms of sine and cosine.

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

4. Cofunction Identities:

These relate trigonometric functions of an angle to trigonometric functions of its complement (90° - θ or π/2 - θ). These are particularly useful when dealing with angles in different quadrants.

• sin(90° - θ) = cos θ
• cos(90° - θ) = sin θ
• tan(90° - θ) = cot θ
• cot(90° - θ) = tan θ
• sec(90° - θ) = csc θ
• csc(90° - θ) = sec θ

Using radians:

• sin(π/2 - θ) = cos θ
• cos(π/2 - θ) = sin θ
• tan(π/2 - θ) = cot θ
• cot(π/2 - θ) = tan θ
• sec(π/2 - θ) = csc θ
• csc(π/2 - θ) = sec θ

5. Identities for Negative Angles (Even and Odd Functions):

These describe how trigonometric functions behave when the angle is negative.

• Sine, Tangent, Cosecant, and Cotangent are odd functions:
  • sin(-θ) = -sin θ
  • tan(-θ) = -tan θ
  • csc(-θ) = -csc θ
  • cot(-θ) = -cot θ
• Cosine and Secant are even functions:
  • cos(-θ) = cos θ
  • sec(-θ) = sec θ

Why are they called even/odd? An even function f(x) satisfies f(-x) = f(x) (like y = x²). An odd function f(x) satisfies f(-x) = -f(x) (like y = x³).

6. Sum and Difference Identities (Brief Mention):

While we won't go into exhaustive detail here, it's important to know that there are identities for the sum and difference of angles, such as:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

These are crucial for more advanced problems and will be covered in more detail if needed.

Example using Identities:

If cos θ = 3/5, find the values of sin θ and tan θ, assuming θ is in the first quadrant.

We use the Pythagorean identity: sin² θ + cos² θ = 1.

sin² θ + (3/5)² = 1

sin² θ + 9/25 = 1

sin² θ = 1 - 9/25 = 16/25

sin θ = ±√(16/25) = ±4/5

Since θ is in the first quadrant, all trigonometric ratios are positive. So, sin θ = 4/5.

Now, we use the quotient identity: tan θ = sin θ / cos θ.

tan θ = (4/5) / (3/5) = (4/5) * (5/3) = 4/3.

Values of Trigonometric Ratios for Standard Angles

It is extremely important to memorize the trigonometric ratios for standard angles like 0°, 30°, 45°, 60°, and 90°.

Angle (θ)	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0° (0 rad)	0	1	0	Undefined	1	Undefined
30° (π/6 rad)	1/2	√3/2	1/√3	2	2/√3	√3
45° (π/4 rad)	1/√2	1/√2	1	√2	√2	1
60° (π/3 rad)	√3/2	1/2	√3	2/√3	2	1/√3
90° (π/2 rad)	1	0	Undefined	1	Undefined	0

Mastering these concepts – the definitions of trigonometric ratios, the conversion between degrees and radians, and the fundamental identities – will provide a strong foundation for tackling all trigonometry-related questions in your examination. Practice these formulas and their applications regularly.