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What is cot (90° - A) = tan A?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The identity cot (90° - A) = tan A tells us that the cotangent of an angle (90° - A) is equal to the tangent of angle A. This is a fundamental trigonometric identity for complementary angles, meaning angles that add up to 90 degrees.
Simple Example
Quick Example
Imagine you are flying a kite. If the angle the kite string makes with the ground is A, then the angle the string makes with a vertical line from the kite to the ground is (90° - A). This identity helps us find relationships between these angles. For instance, if tan A is 1, then cot (90° - A) is also 1.
Worked Example
Step-by-Step
Let's verify cot (90° - A) = tan A for A = 30°.
Step 1: Substitute A = 30° into the expression cot (90° - A).
---Step 2: Calculate the angle inside the cotangent function: 90° - 30° = 60°.
---Step 3: So, we need to find cot 60°.
---Step 4: We know that cot 60° = 1 / tan 60°.
---Step 5: We also know that tan 60° = sqrt(3).
---Step 6: Therefore, cot 60° = 1 / sqrt(3).
---Step 7: Now, let's find tan A for A = 30°. So, tan 30°.
---Step 8: We know that tan 30° = 1 / sqrt(3).
---Step 9: Since cot (90° - 30°) = 1 / sqrt(3) and tan 30° = 1 / sqrt(3), we can see that cot (90° - A) = tan A is true for A = 30°.
Answer: The identity holds true.
Why It Matters
Understanding this identity is crucial in fields like Engineering and Physics, especially when dealing with angles and forces. Engineers use it to design structures, and physicists apply it in wave analysis. It's also foundational for advanced AI/ML algorithms that process spatial data.
Common Mistakes
MISTAKE: Confusing cot (90° - A) with cot A | CORRECTION: Remember that (90° - A) is a different angle from A. The identity specifically relates the cotangent of the complementary angle to the tangent of the original angle.
MISTAKE: Mixing up cot and tan, writing tan (90° - A) = cot A | CORRECTION: While tan (90° - A) = cot A is also true, this specific identity is cot (90° - A) = tan A. Keep them distinct.
MISTAKE: Forgetting the '90° - A' part and just thinking cot A = tan A | CORRECTION: The '90° - A' is key. cot A is generally not equal to tan A. They are only equal for specific angles (like A = 45°).
Practice Questions
Try It Yourself
QUESTION: If tan A = 2, what is the value of cot (90° - A)? | ANSWER: 2
QUESTION: Evaluate cot (90° - 45°). | ANSWER: tan 45° = 1
QUESTION: If cot (90° - A) = sqrt(3), find the value of angle A. | ANSWER: A = 60°
MCQ
Quick Quiz
Which of the following is equivalent to cot (90° - 50°)?
cot 50°
tan 50°
cot 40°
tan 40°
The Correct Answer Is:
B
According to the identity, cot (90° - A) = tan A. Here, A = 50°. So, cot (90° - 50°) = tan 50°. Option C and D are incorrect because cot 40° is not tan 50° and tan 40° is not tan 50°.
Real World Connection
In the Real World
This concept is used in GPS navigation systems in your mobile phone. When satellites send signals, their angles relative to the receiver are crucial. Trigonometric identities help calculate precise positions, much like how ISRO scientists track satellites in space.
Key Vocabulary
Key Terms
Trigonometry: branch of mathematics dealing with angles and triangles | Cotangent (cot): a trigonometric ratio, the reciprocal of tangent | Tangent (tan): a trigonometric ratio, opposite side / adjacent side | Complementary Angles: two angles that add up to 90 degrees | Identity: an equation that is true for all values of the variables
What's Next
What to Learn Next
Next, you should explore other trigonometric identities for complementary angles, like sin (90° - A) = cos A and sec (90° - A) = cosec A. These build on what you've learned and will unlock more complex problem-solving in trigonometry.
