If sin⁡α+cos⁡α=−72, then α is the angle of which quadrant? My attempts: 2sin⁡(π4+α)=−72 sin⁡(π4+α)=−144

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If sin⁡α+cos⁡α=−72, then α is the angle of which quadrant?  My attempts:   2sin⁡(π4+α)=−72   sin⁡(π4+α)=−144

Cassandra Ramirez · Accepted Answer

sin⁡(π4+α)=−144   π4+α=2kπ−sin−1(144)  OR  π4+α=2kπ−π+sin−1(144)    α=(8k−1)π4−sin−1(144)  OR  α=(8k−5)π4+sin−1(144)    α∈{(8k−1)π4−sin−1(144)}∪{(8k−5)π4+sin−1(144)}    Where, k is any integer i.e. k=0,±1,±2,±3,…

Jacob Homer · Accepted Answer

The minimum value of both sin⁡α  and  cos⁡α is −1.   Hence, if either sin⁡α  and  cos⁡α was positive, they could not sum to a value less than −1.   But they sum to −72, which is less than −1.   Hence, both sin⁡α  and  cos⁡α are negative, which means that α is in the 3rd quadrant.

alenahelenash · Accepted Answer

Generally sin is negative from (2n−1)π to 2nπ. So we have: (2n−1)π&amp;lt;α+π4&amp;lt;2nπ ⇒(2n−1)π−π4&amp;lt;α&amp;lt;2nπ−π4 Now you can put different integer values of n to get different ranges for α

If \sin \alpha+\cos \alpha=-\frac{\sqrt 7}{2}, then \alpha is the angle

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