Chapter 6: Trigonometry

Solve · page 105: 1. Find in which quadrant the following angles lie. Write a co-terminal angle for each:

(i) $65^{\circ}$

Solution $$65^{\circ}$$ The angle $65^{\circ}$ lies in the First Quadrant [پہلا ربﻊ] and a co-terminal angle [ہم منتہا زاويہ] is $65^{\circ} + 360^{\circ} = 425^{\circ}$.

Explanation

• Identified the quadrant by noting that the positive angle $65^{\circ}$ is between $0^{\circ}$ and $90^{\circ}$.
• Found the co-terminal angle by adding $360^{\circ}$ to the given angle.

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(ii) $-135^{\circ}$

Solution $$-135^{\circ}$$ $$-135^{\circ} + 360^{\circ} = 225^{\circ}$$ The angle $-135^{\circ}$ lies in the Third Quadrant [تیسرا ربﻊ] and a co-terminal angle [ہم منتہا زاويہ] is $225^{\circ}$.

Explanation

• Added $360^{\circ}$ to the negative angle to find a positive angle with the same terminal side.
• Identified the quadrant by noting that the resulting angle $225^{\circ}$ lies between $180^{\circ}$ and $270^{\circ}$.

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(iii) $-40^{\circ}$

Solution $$-40^{\circ}$$ $$-40^{\circ} + 360^{\circ} = 320^{\circ}$$ The angle $-40^{\circ}$ lies in the Fourth Quadrant [چوتھا ربﻊ] and a co-terminal angle [ہم منتہا زاويہ] is $320^{\circ}$.

Explanation

• Added $360^{\circ}$ to the negative angle to find its equivalent positive angle representation.
• Identified the quadrant by noting that the resulting angle $320^{\circ}$ lies between $270^{\circ}$ and $360^{\circ}$.

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(iv) $210^{\circ}$

Solution $$210^{\circ}$$ The angle $210^{\circ}$ lies in the Third Quadrant [تیسرا ربﻊ] and a co-terminal angle [ہم منتہا زاويہ] is $210^{\circ} + 360^{\circ} = 570^{\circ}$.

Explanation

• Identified the quadrant by noting that the positive angle $210^{\circ}$ is between $180^{\circ}$ and $270^{\circ}$.
• Found the co-terminal angle by adding $360^{\circ}$ to the given angle.

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(v) $-150^{\circ}$

Solution $$-150^{\circ}$$ $$-150^{\circ} + 360^{\circ} = 210^{\circ}$$ The angle $-150^{\circ}$ lies in the Third Quadrant [تیسرا ربﻊ] and a co-terminal angle [ہم منتہا زاويہ] is $210^{\circ}$.

Explanation

• Added $360^{\circ}$ to the negative angle to obtain a positive angle having the same terminal side.
• Identified the quadrant by noting that the resulting angle $210^{\circ}$ lies between $180^{\circ}$ and $270^{\circ}$.