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What is Trigonometric Ratios for (180° - A)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
Trigonometric ratios for (180° - A) tell us how sine, cosine, and tangent values change when an angle is in the second quadrant. It helps us find the trigonometric values of angles larger than 90 degrees by relating them back to acute angles (angles less than 90 degrees).
Simple Example
Quick Example
Imagine you are flying a kite. If the kite string makes an angle of 120 degrees with the ground, you can use the (180° - A) concept to find its sine or cosine. You would relate 120 degrees to (180° - 60°), so the trigonometric ratios of 120 degrees will be similar to those of 60 degrees, with some sign changes.
Worked Example
Step-by-Step
Let's find the value of sin(150°).
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Step 1: Identify the angle and relate it to (180° - A). Here, the angle is 150°.
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Step 2: Find 'A' such that 180° - A = 150°. So, A = 180° - 150° = 30°.
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Step 3: Apply the rule for sin(180° - A), which is sin(A). So, sin(150°) = sin(30°).
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Step 4: Recall the value of sin(30°).
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Step 5: sin(30°) = 1/2.
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Therefore, sin(150°) = 1/2. The sign remains positive because sine is positive in the second quadrant.
Why It Matters
Understanding these ratios is crucial in fields like Physics to calculate forces and trajectories, and in Engineering for designing structures. Even Space Technology uses these concepts for satellite navigation and rocket launches, helping engineers at ISRO map out complex paths.
Common Mistakes
MISTAKE: Forgetting the sign change for cosine and tangent. Students often assume all ratios remain positive. | CORRECTION: Remember that in the second quadrant (angles between 90° and 180°), only sine is positive. Cosine and tangent are negative.
MISTAKE: Confusing (180° - A) with (90° + A) or other identities. | CORRECTION: The (180° - A) identity keeps the trigonometric function (sin remains sin, cos remains cos). The angle 'A' is always acute.
MISTAKE: Calculating 'A' incorrectly, leading to wrong acute angles. For example, for 135 degrees, using 180 - 45 = 135, but then using 45 degrees for the ratio. | CORRECTION: 'A' is the reference angle. Make sure A = 180° - given angle.
Practice Questions
Try It Yourself
QUESTION: Find the value of cos(120°). | ANSWER: -1/2
QUESTION: If tan(A) = 1, what is the value of tan(180° - A)? | ANSWER: -1
QUESTION: Given that sin(X) = 0.8 and X is an acute angle, find the value of sin(180° - X) + cos(180° - X). (Hint: You might need Pythagoras for cos(X)). | ANSWER: 0.8 - 0.6 = 0.2 (assuming cos(X) = 0.6)
MCQ
Quick Quiz
Which of the following statements is true for an angle A, where 0° < A < 90°?
sin(180° - A) = -sin(A)
cos(180° - A) = cos(A)
tan(180° - A) = -tan(A)
All of the above
The Correct Answer Is:
C
In the second quadrant (180° - A), sine is positive, while cosine and tangent are negative. So, sin(180° - A) = sin(A), cos(180° - A) = -cos(A), and tan(180° - A) = -tan(A).
Real World Connection
In the Real World
These ratios are used by architects to calculate the stability of buildings, especially for roof angles or ramp slopes. For example, when designing a flyover in Mumbai, engineers use trigonometry to ensure the ramp's angle is safe and efficient for vehicles, often dealing with angles greater than 90 degrees when considering different parts of the structure.
Key Vocabulary
Key Terms
QUADRANT: One of four sections of a coordinate plane | ACUTE ANGLE: An angle less than 90 degrees | REFERENCE ANGLE: The acute angle between the terminal arm of an angle and the x-axis | TRIGONOMETRIC RATIOS: Ratios of sides of a right-angled triangle (sine, cosine, tangent)
What's Next
What to Learn Next
Next, you should explore trigonometric ratios for (90° + A) and (360° - A). These concepts will further expand your ability to work with angles in all four quadrants, which is essential for advanced trigonometry and its applications.
