What is the Derivation of Trigonometric Ratios for 75 Degrees?

S6-SA2-0376

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The derivation of trigonometric ratios for 75 degrees involves finding the values of sin 75, cos 75, and tan 75 using combinations of standard angles like 30, 45, and 60 degrees. We use trigonometric sum or difference formulas to break down 75 degrees into angles whose ratios we already know. This helps us calculate exact values for these less common angles.

Simple Example

Quick Example

Imagine you need to know the exact 'score' of sin 75 degrees, but your math 'scorecard' only has sin 30, sin 45, and sin 60. Deriving sin 75 is like figuring out that sin 75 is the 'combined score' of sin (45 + 30), allowing you to use known values to get the unknown one. It's like finding the price of a special chai blend by knowing the prices of its individual ingredients.

Worked Example

Step-by-Step

Let's find the value of sin 75 degrees.

Step 1: Express 75 degrees as a sum of standard angles. We know 75 = 45 + 30.

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Step 2: Use the trigonometric sum formula for sine: sin(A + B) = sin A cos B + cos A sin B.

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Step 3: Substitute A = 45 degrees and B = 30 degrees into the formula.
sin(75) = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30.

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Step 4: Recall the standard trigonometric values:
sin 45 = 1/sqrt(2)
cos 30 = sqrt(3)/2
cos 45 = 1/sqrt(2)
sin 30 = 1/2

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Step 5: Substitute these values into the equation.
sin(75) = (1/sqrt(2)) * (sqrt(3)/2) + (1/sqrt(2)) * (1/2)

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Step 6: Simplify the expression.
sin(75) = sqrt(3)/(2*sqrt(2)) + 1/(2*sqrt(2))
sin(75) = (sqrt(3) + 1) / (2*sqrt(2))

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Step 7: Rationalize the denominator by multiplying the numerator and denominator by sqrt(2).
sin(75) = ((sqrt(3) + 1) * sqrt(2)) / ((2*sqrt(2)) * sqrt(2))
sin(75) = (sqrt(6) + sqrt(2)) / (2 * 2)
sin(75) = (sqrt(6) + sqrt(2)) / 4

Answer: The value of sin 75 degrees is (sqrt(6) + sqrt(2)) / 4.

Why It Matters

Knowing how to derive these ratios is crucial for engineers designing structures or calculating projectile paths in Physics. In AI/ML, these concepts help in understanding signal processing and image recognition algorithms. Doctors use similar principles in medical imaging, and space scientists at ISRO use trigonometry for satellite trajectories.

Common Mistakes

MISTAKE: Assuming sin(A + B) = sin A + sin B | CORRECTION: Remember the sum formula sin(A + B) = sin A cos B + cos A sin B. Trigonometric functions are not distributive like simple multiplication.

MISTAKE: Forgetting to rationalize the denominator or making errors while doing so | CORRECTION: Always check if your final answer has an irrational number in the denominator and multiply both numerator and denominator by the irrational term to simplify.

MISTAKE: Using incorrect standard angle values (e.g., mixing up sin 30 and cos 60) | CORRECTION: Memorize the trigonometric table for 0, 30, 45, 60, and 90 degrees accurately. A quick sketch of a right-angled triangle can help verify.

Practice Questions

Try It Yourself

QUESTION: Derive the value of cos 75 degrees. | ANSWER: (sqrt(6) - sqrt(2)) / 4

QUESTION: Derive the value of tan 75 degrees. | ANSWER: 2 + sqrt(3)

QUESTION: Using the values of sin 75 and cos 75, verify that sin^2(75) + cos^2(75) = 1. | ANSWER: ( (sqrt(6)+sqrt(2))/4 )^2 + ( (sqrt(6)-sqrt(2))/4 )^2 = (6+2+2sqrt(12))/16 + (6+2-2sqrt(12))/16 = (8+4sqrt(3))/16 + (8-4sqrt(3))/16 = (8+4sqrt(3)+8-4sqrt(3))/16 = 16/16 = 1. Verified.

MCQ

Quick Quiz

Which of the following expressions correctly represents sin 75 degrees?

sin 45 + sin 30

sin 45 cos 30 + cos 45 sin 30

cos 45 cos 30 - sin 45 sin 30

sin 60 + sin 15

The Correct Answer Is:

Option B correctly uses the sum formula sin(A + B) = sin A cos B + cos A sin B, with A=45 and B=30. Options A and D are incorrect applications of the formula, and Option C is the formula for cos(A+B).

Real World Connection

In the Real World

Imagine an architect in Mumbai designing a slanted roof or a ramp. If the desired angle is 75 degrees, they need to know the exact sine or cosine value to calculate the height or length of materials needed. Similarly, a cricket analyst might use these ratios to model the trajectory of a shot hit at a specific angle to predict where it will land.

Key Vocabulary

Key Terms

DERIVATION: The process of obtaining a mathematical result from other known results or principles | TRIGONOMETRIC RATIOS: Ratios of sides of a right-angled triangle related to its angles (sine, cosine, tangent) | STANDARD ANGLES: Common angles like 0, 30, 45, 60, 90 degrees whose trigonometric ratios are well-known | SUM FORMULA: A trigonometric identity used to find the ratio of the sum of two angles, e.g., sin(A+B) | RATIONALIZE: To remove irrational numbers (like square roots) from the denominator of a fraction.

What's Next

What to Learn Next

Next, you can explore the derivation of trigonometric ratios for 15 degrees and 105 degrees, which also use similar sum and difference formulas. Understanding these will strengthen your grasp of trigonometric identities and prepare you for more complex problems in higher classes.