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What is cosec (90° - A) = sec A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity cosec (90° - A) = sec A means that the cosecant of an angle (90° - A) is equal to the secant of the angle A. This is one of the important trigonometric identities for complementary angles, showing a special relationship between trigonometric ratios.
Simple Example
Quick Example
Imagine you're watching a cricket match and the commentator says, 'The angle of elevation for that shot was 30 degrees.' If you wanted to find cosec (90° - 30°), this identity tells you it's the same as sec 30°. So, cosec 60° = sec 30°.
Worked Example
Step-by-Step
Let's verify cosec (90° - 30°) = sec 30°.
---Step 1: Identify the angle A. Here, A = 30°.
---Step 2: Calculate (90° - A). So, (90° - 30°) = 60°.
---Step 3: Find the value of cosec (90° - A), which is cosec 60°. We know that sin 60° = sqrt(3)/2, and cosec 60° = 1/sin 60° = 1 / (sqrt(3)/2) = 2/sqrt(3).
---Step 4: Find the value of sec A, which is sec 30°. We know that cos 30° = sqrt(3)/2, and sec 30° = 1/cos 30° = 1 / (sqrt(3)/2) = 2/sqrt(3).
---Step 5: Compare the results. Both cosec 60° and sec 30° are equal to 2/sqrt(3).
---Answer: Thus, cosec (90° - 30°) = sec 30° is verified.
Why It Matters
Understanding these identities is crucial for solving complex problems in physics, like calculating projectile motion or wave patterns. Engineers use these concepts in designing structures and circuits, while data scientists in AI/ML might use them in algorithms for signal processing. It opens doors to careers in ISRO, DRDO, or even developing new mobile apps.
Common Mistakes
MISTAKE: Confusing cosec (90° - A) with cosec A or sec (90° - A). | CORRECTION: Remember the specific pair: cosec and sec are complementary for (90° - A). Similarly, sin and cos, tan and cot.
MISTAKE: Incorrectly applying the identity to angles that are not (90° - A). For example, thinking cosec (180° - A) = sec A. | CORRECTION: This identity only works for complementary angles, specifically (90° - A). For other angles, different identities apply.
MISTAKE: Forgetting the definitions of cosec and sec (i.e., 1/sin and 1/cos). | CORRECTION: Always recall that cosec A = 1/sin A and sec A = 1/cos A. This helps in deriving and understanding the identities.
Practice Questions
Try It Yourself
QUESTION: If A = 45°, what is the value of cosec (90° - A)? | ANSWER: cosec (90° - 45°) = cosec 45° = sqrt(2)
QUESTION: If cosec (90° - X) = 2, find the value of cos X. | ANSWER: cosec (90° - X) = sec X. So, sec X = 2. This means cos X = 1/sec X = 1/2.
QUESTION: Prove that cosec (90° - 20°) - sec 20° = 0. | ANSWER: We know cosec (90° - A) = sec A. So, cosec (90° - 20°) = sec 20°. Therefore, sec 20° - sec 20° = 0. Hence proved.
MCQ
Quick Quiz
Which of the following is equal to cosec (90° - 50°)?
sin 50°
cos 50°
sec 50°
tan 50°
The Correct Answer Is:
C
The identity states that cosec (90° - A) = sec A. Here, A = 50°, so cosec (90° - 50°) = sec 50°. Options A, B, and D are incorrect as they represent different trigonometric ratios.
Real World Connection
In the Real World
These identities are used by architects in India to calculate angles for building designs, ensuring stability and aesthetics. For example, when designing a sloped roof or a ramp, they use trigonometry to determine the correct rise and run, and these identities help simplify calculations involving complementary angles. Even ISRO scientists use these principles for satellite trajectory calculations.
Key Vocabulary
Key Terms
COMPLEMENTARY ANGLES: Two angles whose sum is 90° | COSECANT (cosec): The reciprocal of the sine function (1/sin) | SECANT (sec): The reciprocal of the cosine function (1/cos) | TRIGONOMETRIC IDENTITY: An equation involving trigonometric ratios that is true for all values of the angle | RATIO: A comparison of two numbers by division
What's Next
What to Learn Next
Next, you should explore other trigonometric identities for complementary angles, like sin (90° - A) = cos A and tan (90° - A) = cot A. This will build a complete picture of how these angles relate and help you solve more complex problems in trigonometry.
