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What is cos (90° - A) = sin A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity cos (90° - A) = sin A tells us that the cosine of an angle (90° - A) is equal to the sine of the angle A. This relationship is true for all acute angles A in a right-angled triangle, showing how sine and cosine are related as 'cofunctions'.
Simple Example
Quick Example
Imagine you're looking at a right-angled triangle. If one acute angle is 30 degrees, then the other acute angle must be (90 - 30) = 60 degrees. This identity says that if you find cos(60°), it will be the same value as sin(30°).
Worked Example
Step-by-Step
Let's verify cos (90° - A) = sin A for A = 30°.
---Step 1: Substitute A = 30° into the left side of the equation: cos (90° - 30°).
---Step 2: Calculate the angle inside the parenthesis: cos (60°).
---Step 3: Recall the value of cos (60°) from the trigonometric table. cos (60°) = 1/2.
---Step 4: Now, substitute A = 30° into the right side of the equation: sin (30°).
---Step 5: Recall the value of sin (30°) from the trigonometric table. sin (30°) = 1/2.
---Step 6: Compare the results from Step 3 and Step 5. Since 1/2 = 1/2, the identity is verified.
Answer: cos (90° - 30°) = sin (30°) is true.
Why It Matters
Understanding this relationship is crucial in many advanced fields like AI/ML for signal processing, in Physics for analyzing wave motions, and in Engineering for designing structures. Engineers use these concepts to calculate forces and angles, ensuring bridges and buildings are safe and stable, impacting millions of lives daily.
Common Mistakes
MISTAKE: Thinking cos (90° - A) = -sin A | CORRECTION: Remember that for acute angles, all trigonometric ratios are positive. The identity is cos (90° - A) = sin A.
MISTAKE: Confusing the angle inside the bracket, e.g., writing cos (A - 90°) = sin A | CORRECTION: The identity specifically uses (90° - A). The order of subtraction matters.
MISTAKE: Applying this identity to angles that are not acute without checking the quadrant rules | CORRECTION: While the core relationship holds, for angles outside 0-90°, the signs might change based on the quadrant. For Class 10, focus on acute angles.
Practice Questions
Try It Yourself
QUESTION: If sin (50°) = 0.766, what is the value of cos (40°)? | ANSWER: 0.766
QUESTION: Simplify the expression: cos (90° - 25°) + sin (25°). | ANSWER: 2 sin (25°)
QUESTION: If cos (90° - x) = 0.8, find the value of sin x. Then, using Pythagoras identity, find cos x. | ANSWER: sin x = 0.8, cos x = 0.6
MCQ
Quick Quiz
Which of the following is equivalent to sin (70°)?
cos (70°)
cos (20°)
sin (20°)
tan (20°)
The Correct Answer Is:
B
The identity states that sin A = cos (90° - A). So, sin (70°) = cos (90° - 70°) = cos (20°).
Real World Connection
In the Real World
This concept is used in GPS systems that help auto-rickshaw drivers find the shortest routes. Satellites use trigonometry to calculate positions and distances, relying on angle relationships like these to pinpoint locations accurately, making navigation apps like Google Maps work seamlessly across India.
Key Vocabulary
Key Terms
TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles | SINE (sin): A trigonometric ratio defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle | COSINE (cos): A trigonometric ratio defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle | ACUTE ANGLE: An angle that measures less than 90 degrees | COMPLEMENTARY ANGLES: Two angles whose sum is 90 degrees.
What's Next
What to Learn Next
Next, you should explore other trigonometric identities, especially those involving tan, cosec, sec, and cot. Understanding these will help you solve more complex problems and build a strong foundation for higher mathematics.
