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What is tan (90° - A) = cot A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity tan (90° - A) = cot A shows a special relationship between tangent and cotangent functions for complementary angles. It means that the tangent of an angle is equal to the cotangent of its complementary angle (90° minus that angle). This is a fundamental trigonometric identity used to simplify expressions and solve problems.
Simple Example
Quick Example
Imagine you are flying a kite. If the angle the kite string makes with the ground is 30°, then the angle it makes with an imaginary vertical line going straight up is 90° - 30° = 60°. This identity tells us that tan 30° will have the same value as cot 60°. It's like saying the price of a 'chai' is the same as the price of a 'coffee' if they are complementary drinks!
Worked Example
Step-by-Step
Let's find the value of tan 40° if we know cot 50° = 0.8391.
Step 1: Identify the given information. We know cot 50° = 0.8391.
---Step 2: Recognize the relationship between 40° and 50°. They are complementary angles because 40° + 50° = 90°.
---Step 3: Apply the identity tan (90° - A) = cot A. Here, A = 40°.
---Step 4: So, tan 40° = cot (90° - 40°).
---Step 5: Simplify the expression: tan 40° = cot 50°.
---Step 6: Substitute the known value: tan 40° = 0.8391.
Answer: tan 40° = 0.8391.
Why It Matters
This identity is crucial in fields like Engineering and Physics for calculating angles and distances, such as designing bridges or understanding light refraction. In Space Technology, it helps ISRO scientists track satellites or plan rocket trajectories by simplifying complex angular calculations. It's a basic tool for anyone working with shapes and angles, from architects to game developers.
Common Mistakes
MISTAKE: Writing tan (90° + A) = cot A | CORRECTION: The identity is for (90° - A), meaning complementary angles. (90° + A) is a different concept related to angles in other quadrants.
MISTAKE: Confusing tan (90° - A) with sin (90° - A) or cos (90° - A) | CORRECTION: Remember that tan (90° - A) = cot A, while sin (90° - A) = cos A and cos (90° - A) = sin A. Each pair has its own complementary relationship.
MISTAKE: Assuming tan (90° - A) = 1 / tan A | CORRECTION: While cot A = 1 / tan A, the identity is tan (90° - A) = cot A directly, not 1 / tan A. It's about complementary angles, not just reciprocals of the same angle.
Practice Questions
Try It Yourself
QUESTION: If tan 25° = 0.4663, what is the value of cot 65°? | ANSWER: cot 65° = 0.4663
QUESTION: Simplify the expression: tan 70° / cot 20°. | ANSWER: 1
QUESTION: If tan (3A) = cot (A - 20°), where 3A is an acute angle, find the value of A. | ANSWER: A = 27.5°
MCQ
Quick Quiz
Which of the following is equal to tan 15°?
cot 15°
cot 75°
tan 75°
1 / tan 15°
The Correct Answer Is:
B
According to the identity tan (90° - A) = cot A, tan 15° = tan (90° - 75°) = cot 75°. Option D is the reciprocal identity, not the complementary angle identity.
Real World Connection
In the Real World
This concept is used by civil engineers in India when designing the slopes of roads or ramps. If they know the angle of inclination of a ramp (say, 30°), they can quickly find its complementary angle (60°) to calculate related forces or dimensions using trigonometric ratios, ensuring the structure is safe and stable.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all values of the variables involved | COMPLEMENTARY ANGLES: Two angles whose sum is 90° | TANGENT (tan): A trigonometric ratio, opposite side / adjacent side | COTANGENT (cot): A trigonometric ratio, adjacent side / opposite side (reciprocal of tangent)
What's Next
What to Learn Next
Next, you should explore other trigonometric identities for complementary angles, such as sin (90° - A) = cos A and sec (90° - A) = cosec A. Understanding these will give you a complete picture of how trigonometry works with angles.
