What is sin (90° - A) = cos A? | Simple Explanation for Class 10

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What is sin (90° - A) = cos A?

Grade Level:

Class 10

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

Definition

What is it?

The identity sin (90° - A) = cos A shows a special relationship between the sine and cosine of complementary angles. It means that the sine of an angle is equal to the cosine of its complementary angle (the angle that adds up to 90 degrees with it). This is a fundamental concept in trigonometry.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, whose ages add up to 90 years. If Rahul's age is 'A', then Priya's age is '90 - A'. This identity is like saying that the 'sin value' of Rahul's age is the same as the 'cos value' of Priya's age. For example, sin 30 degrees is 0.5, and cos (90 - 30) degrees, which is cos 60 degrees, is also 0.5.

Worked Example

Step-by-Step

PROBLEM: If sin 40 degrees = 0.64, what is the value of cos 50 degrees?

STEP 1: Identify the given information. We know sin 40 degrees = 0.64.
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STEP 2: Recognize the relationship between the angles. We need to find cos 50 degrees. Notice that 40 degrees + 50 degrees = 90 degrees. This means 40 degrees and 50 degrees are complementary angles.
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STEP 3: Apply the identity sin (90° - A) = cos A. Here, if A = 40 degrees, then (90° - A) = (90° - 40°) = 50 degrees.
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STEP 4: Substitute the values into the identity. So, sin 40 degrees = cos (90° - 40°) = cos 50 degrees.
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STEP 5: Since sin 40 degrees is given as 0.64, then cos 50 degrees must also be 0.64.

ANSWER: cos 50 degrees = 0.64.

Why It Matters

This identity is crucial for simplifying complex calculations in fields like Physics and Engineering, especially when dealing with waves or forces. Engineers use this to design structures, and scientists in Space Technology apply it to calculate satellite trajectories. Learning this helps you build a strong base for future careers in AI/ML or even Medicine, where precise measurements are key.

Common Mistakes

MISTAKE: Thinking sin A = cos (90° + A) | CORRECTION: The relationship is for complementary angles, meaning they add up to 90 degrees, so it's sin A = cos (90° - A).

MISTAKE: Confusing sin with tan or sec. For example, writing tan (90° - A) = cot A as sin (90° - A) = cot A | CORRECTION: Each trigonometric ratio has its specific complementary identity. For sine and cosine, it's sin (90° - A) = cos A.

MISTAKE: Forgetting that 'A' represents an angle in degrees (or radians, but usually degrees at this level). For example, thinking A is just a number. | CORRECTION: 'A' is always an angle measurement, like 30 degrees, 45 degrees, etc., and the 'sin' or 'cos' operation is performed on that angle.

Practice Questions

Try It Yourself

QUESTION: If cos 70 degrees = 0.34, what is sin 20 degrees? | ANSWER: sin 20 degrees = 0.34

QUESTION: Simplify the expression: sin 65 degrees / cos 25 degrees. | ANSWER: 1

QUESTION: If sin (3x - 10 degrees) = cos (x + 20 degrees), find the value of x. Assume 3x - 10 and x + 20 are acute angles. | ANSWER: x = 20 degrees

MCQ

Quick Quiz

Which of the following is equivalent to cos 35 degrees?

sin 35 degrees

sin 55 degrees

cos 55 degrees

sin (90 + 35) degrees

The Correct Answer Is:

According to the identity, cos A = sin (90° - A). So, cos 35 degrees = sin (90° - 35°) = sin 55 degrees. Option B is correct because 35 and 55 are complementary angles.

Real World Connection

In the Real World

When ISRO launches a rocket, they use trigonometry to calculate its path and angle of ascent. The relationship between sine and cosine of complementary angles helps engineers quickly check calculations or simplify formulas related to the rocket's trajectory or the angles of solar panels on a satellite. It ensures precision, just like how UPI transactions need precise calculations!

Key Vocabulary

Key Terms

TRIGONOMETRY: The branch of mathematics dealing with the relationships between the sides and angles of triangles. | SINE: A trigonometric ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. | COSINE: A trigonometric ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. | COMPLEMENTARY ANGLES: Two angles whose sum is 90 degrees. | IDENTITY: An equation that is true for all possible values of its variables.

What's Next

What to Learn Next

Great job understanding this! Next, you should explore other trigonometric identities for complementary angles, like tan (90° - A) = cot A and sec (90° - A) = cosec A. These build directly on what you've learned and will help you solve even more complex problems in trigonometry.