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What is sec (90° - A) = cosec A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity sec (90° - A) = cosec A shows a relationship between two trigonometric ratios. It means that the secant of an angle (90° - A) is always equal to the cosecant of the angle A. This is part of understanding complementary angles in trigonometry.
Simple Example
Quick Example
Imagine you are looking at a right-angled triangle. If one acute angle is 30 degrees, then the other acute angle is 90 - 30 = 60 degrees. This identity tells us that sec(60°) will have the same value as cosec(30°). It's like saying the cost of a 1 kg bag of rice is the same as the cost of 10 small packets of salt!
Worked Example
Step-by-Step
Let's verify sec (90° - 30°) = cosec 30°.
Step 1: Calculate the angle (90° - A). Here, A = 30°, so 90° - 30° = 60°.
---Step 2: Find the value of sec (90° - A), which is sec 60°. We know that sec x = 1/cos x. So, sec 60° = 1/cos 60°.
---Step 3: From the trigonometric table, cos 60° = 1/2. So, sec 60° = 1 / (1/2) = 2.
---Step 4: Now, find the value of cosec A, which is cosec 30°. We know that cosec x = 1/sin x. So, cosec 30° = 1/sin 30°.
---Step 5: From the trigonometric table, sin 30° = 1/2. So, cosec 30° = 1 / (1/2) = 2.
---Step 6: Compare the results. sec (90° - 30°) = 2 and cosec 30° = 2. Since both are equal, the identity is verified.
Answer: sec (90° - 30°) = cosec 30° is verified as 2 = 2.
Why It Matters
Understanding these trigonometric identities is crucial for fields like Physics and Engineering, especially when dealing with waves, signals, or forces. Engineers use these concepts to design everything from bridges to mobile phone antennas. Even in Space Technology, calculating rocket trajectories uses these principles to ensure successful missions.
Common Mistakes
MISTAKE: Writing sec (90° - A) = sec A | CORRECTION: Remember it changes to cosec A, not stays as sec A. It's a complementary angle identity.
MISTAKE: Confusing cosec A with cos A | CORRECTION: Cosecant (cosec) is the reciprocal of sine, while cosine (cos) is a different ratio altogether. Always double-check which ratio you are using.
MISTAKE: Applying this identity when the angle is not (90° - A) | CORRECTION: This identity specifically works for (90° - A). For example, sec (180° - A) would be different.
Practice Questions
Try It Yourself
QUESTION: If A = 45°, what is sec (90° - A)? | ANSWER: cosec 45°
QUESTION: Evaluate sec (90° - 60°) using the identity. | ANSWER: cosec 60° = 2/sqrt(3)
QUESTION: If sec (90° - B) = 2, find the value of angle B. | ANSWER: cosec B = 2, so sin B = 1/2. Therefore, B = 30°.
MCQ
Quick Quiz
Which of the following is equivalent to sec (90° - 25°)?
sec 25°
cosec 25°
cos 25°
sin 25°
The Correct Answer Is:
B
According to the complementary angle identity, sec (90° - A) = cosec A. Here A is 25°, so sec (90° - 25°) = cosec 25°. Options A, C, and D are incorrect as they don't follow this rule.
Real World Connection
In the Real World
Imagine an architect designing a new building in Mumbai. They use trigonometry to calculate angles for roofs, ramps, and even sun shades. When they calculate the angle for a sloped roof, they might need to find a related angle for support beams. These identities help them quickly switch between different trigonometric ratios without complex recalculations, ensuring the building is safe and strong.
Key Vocabulary
Key Terms
TRIGONOMETRIC RATIOS: Relationships between the angles and sides of a right-angled triangle, like sine, cosine, tangent | SECANT (sec): The reciprocal of the cosine of an angle (1/cos A) | COSECANT (cosec): The reciprocal of the sine of an angle (1/sin A) | COMPLEMENTARY ANGLES: Two angles whose sum is 90 degrees
What's Next
What to Learn Next
Great job understanding this identity! Next, you should explore other complementary angle identities like sin (90° - A) = cos A and tan (90° - A) = cot A. These will further strengthen your trigonometry basics, making you ready for more advanced problems!
