Trigonometric Functions: A Deep Dive for Class 11 CBSE Maths

Understanding Angles: Degrees and Radians

Angle

An angle is a measure of rotation of a given ray about its initial point. The initial position of the ray is called the initial side, and the final position after rotation is called the terminal side. The point of rotation is called the vertex.

Degree Measure

If a rotation from the initial side to the terminal side is (1/360)th of a revolution, the angle is said to have a measure of one degree (1°). Degrees are further subdivided into minutes (') and seconds ("), where 1° = 60' and 1' = 60".

Radian Measure

The angle subtended at the centre by an arc of length one unit in a unit circle (a circle with radius 1 unit) is said to have a measure of one radian. The circumference of a circle with radius 'r' is 2πr. So, an angle of 2π radians corresponds to a full revolution (360°).

Relation between Degree and Radian

The fundamental relation is π radians = 180°. From this, we can derive 1 radian = 180°/π (approx. 57.295°) and 1° = π/180 radians (approx. 0.01745 radians). This conversion is crucial for various calculations.

Defining Trigonometric Functions with the Unit Circle

In Class 10, trigonometric ratios like sine, cosine, and tangent were defined for acute angles in a right-angled triangle. For Class 11, we extend these definitions to any angle using the unit circle. A unit circle is a circle centered at the origin (0,0) with a radius of 1 unit.

Imagine a point P(x, y) on the unit circle. Let θ be the angle (in radians or degrees) that the line segment OP makes with the positive x-axis, measured anti-clockwise. Then, we define the six trigonometric functions as follows:

• Sine (sin θ): The y-coordinate of point P, so sin θ = y.
• Cosine (cos θ): The x-coordinate of point P, so cos θ = x.
• Tangent (tan θ): The ratio of the y-coordinate to the x-coordinate, tan θ = y/x = sin θ / cos θ, provided x ≠ 0 (i.e., cos θ ≠ 0).
• Cosecant (csc θ): The reciprocal of sine, csc θ = 1/y = 1/sin θ, provided y ≠ 0 (i.e., sin θ ≠ 0).
• Secant (sec θ): The reciprocal of cosine, sec θ = 1/x = 1/cos θ, provided x ≠ 0 (i.e., cos θ ≠ 0).
• Cotangent (cot θ): The reciprocal of tangent, cot θ = x/y = cos θ / sin θ, provided y ≠ 0 (i.e., sin θ ≠ 0).

This unit circle definition allows us to define trigonometric functions for angles greater than 90°, negative angles, and even angles greater than 360°. The signs of these functions depend on the quadrant in which the terminal side of the angle lies. Remember the "All Silver Tea Cups" (ASTC) rule:

• Quadrant I (0° to 90° or 0 to π/2): All trigonometric functions are positive.
• Quadrant II (90° to 180° or π/2 to π): Sine and its reciprocal (cosecant) are positive.
• Quadrant III (180° to 270° or π to 3π/2): Tangent and its reciprocal (cotangent) are positive.
• Quadrant IV (270° to 360° or 3π/2 to 2π): Cosine and its reciprocal (secant) are positive.

We also study fundamental trigonometric identities like sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ, which are derived directly from the Pythagorean theorem applied to the coordinates (x,y) on the unit circle. These identities are crucial for simplifying expressions and solving trigonometric equations.

Worked Examples on Trigonometric Functions

• Example 1: Convert an angle from degrees to radians and find its trigonometric value. Q: Convert 240° into radian measure. Then, find the value of sin(240°). A: Step 1: Conversion from degrees to radians. We know that 180° = π radians. Therefore, 1° = (π/180) radians. So, 240° = 240 (π/180) radians = (4 60)/(3 60) π radians = 4π/3 radians. Step 2: Finding sin(240°). The angle 240° lies in the third quadrant (between 180° and 270°). In the third quadrant, sine is negative. We can write 240° as (180° + 60°). Using the reduction formula sin(180° + θ) = -sin θ, we have sin(240°) = sin(180° + 60°) = -sin(60°). We know sin(60°) = √3/2. Final Answer: 240° is 4π/3 radians, and sin(240°) = -√3/2.
• Example 2: Determine the quadrant and the value of a trigonometric function given one function and its sign. Q: If cos x = -3/5 and x lies in the third quadrant, find the values of the other five trigonometric functions. A: Step 1: Identify the given information. cos x = -3/5, and x is in the third quadrant. In the third quadrant, only tan x and cot x are positive. All other functions (sin x, csc x, sec x) will be negative. Step 2: Use the identity sin²x + cos²x = 1 to find sin x. sin²x + (-3/5)² = 1 sin²x + 9/25 = 1 sin²x = 1 - 9/25 = 16/25 sin x = ±√(16/25) = ±4/5. Since x is in the third quadrant, sin x must be negative. So, sin x = -4/5. Step 3: Calculate the remaining functions using their definitions. tan x = sin x / cos x = (-4/5) / (-3/5) = 4/3. csc x = 1 / sin x = 1 / (-4/5) = -5/4. sec x = 1 / cos x = 1 / (-3/5) = -5/3. cot x = 1 / tan x = 1 / (4/3) = 3/4. Final Answer: sin x = -4/5, tan x = 4/3, csc x = -5/4, sec x = -5/3, cot x = 3/4.

Exam Tips for Trigonometric Functions

To excel in this chapter, consistent practice is key. Here are some important tips:

• Master the Unit Circle: Visualize the unit circle for angles. It helps in understanding signs of functions in different quadrants and quickly recalling values for standard angles (0°, 30°, 45°, 60°, 90°, and their multiples).
• Memorize Fundamental Identities: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x are your best friends. Know them thoroughly.
• Conversion Accuracy: Be careful while converting between degrees and radians. Remember π radians = 180°. A common mistake is using 180/π for radians to degrees, instead of π/180 for degrees to radians.
• Quadrant Analysis: Always determine the quadrant of the angle first when finding values of trigonometric functions. This will help you correctly assign the positive or negative sign.
• Domain and Range: Be aware of the domain (what angles are allowed) and range (what values the function can take) for each trigonometric function, as some functions are undefined at certain angles (e.g., tan 90°).
• Practice Identities: Solving problems involving trigonometric identities requires practice. Try to convert everything to sine and cosine if you're stuck, as this often reveals simplifications. Use reduction formulas (like sin(π ± x), cos(2π ± x)) effectively.

Practice Questions with Solutions

• Q: Find the radian measure corresponding to 225°. A: Step 1: Use the conversion factor from degrees to radians. We know that 180° = π radians. Therefore, 1° = (π/180) radians. Step 2: Multiply the given degree measure by the conversion factor. 225° = 225 (π/180) radians. Simplify the fraction: 225/180 can be divided by 45 (225 = 5 45, 180 = 4 * 45). 225° = (5π/4) radians. Final answer: The radian measure corresponding to 225° is 5π/4.
• Q: Find the degree measure corresponding to (5π/6) radians. A: Step 1: Use the conversion factor from radians to degrees. We know that π radians = 180°. Therefore, 1 radian = (180/π)°. Step 2: Multiply the given radian measure by the conversion factor. (5π/6) radians = (5π/6) (180/π)°. Cancel out π and simplify the numerical part. (5/6) 180° = 5 (180/6)° = 5 30° = 150°. Final answer: The degree measure corresponding to (5π/6) radians is 150°.
• Q: If cot x = -5/12 and x lies in the second quadrant, find the values of sin x and sec x. A: Step 1: Use the identity 1 + cot²x = csc²x to find csc x. 1 + (-5/12)² = csc²x 1 + 25/144 = csc²x (144 + 25)/144 = csc²x 169/144 = csc²x csc x = ±√(169/144) = ±13/12. Step 2: Determine the sign of csc x. Since x is in the second quadrant, sine is positive, so cosecant must also be positive. Therefore, csc x = 13/12. Step 3: Find sin x from csc x. sin x = 1 / csc x = 1 / (13/12) = 12/13. Step 4: Find tan x from cot x. tan x = 1 / cot x = 1 / (-5/12) = -12/5. Step 5: Use the identity 1 + tan²x = sec²x to find sec x. 1 + (-12/5)² = sec²x 1 + 144/25 = sec²x (25 + 144)/25 = sec²x 169/25 = sec²x sec x = ±√(169/25) = ±13/5. Step 6: Determine the sign of sec x. Since x is in the second quadrant, cosine is negative, so secant must also be negative. Therefore, sec x = -13/5. Final answer: sin x = 12/13 and sec x = -13/5.
• Q: Evaluate cos(3π/2 + x) if sin x = 1/2 and x is in the first quadrant. A: Step 1: Use the reduction formula for cos(3π/2 + x). The angle (3π/2 + x) lies in the fourth quadrant. In the fourth quadrant, cosine is positive. When the angle involves 3π/2 (or 270°), the trigonometric function changes from cosine to sine. So, cos(3π/2 + x) = sin x. Step 2: Substitute the given value of sin x. Given sin x = 1/2. Final answer: cos(3π/2 + x) = 1/2.

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