If SecA + TanA = 3 Find sinA + cos A
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Given: Sec A + Tan A = 3
To find: The value of sin A + cos A
Solution:
- Now we have given that sec A + tan A = 3
- Converting it to sin and cos, we get:
1 / cos A + sin A / cos A = 3
1 + sin A = 3 cos A
- Now squaring on both the sides, we get:
( 1 + sin A )^2 = 9 cos^2 A
1 + sin^2 A + 2 sin A = 9 ( 1 - sin^2 A )
1 + sin^2 A + 2 sin A = 9 - 9 sin^2 A
9 sin^2 A + sin^2 A + 2 sin A = 9 - 1
10 sin^2 A + 2 sin A - 8 = 0
10 sin^2 A + 10 sin A - 8 sin A - 8 = 0
10 sin A (sin A + 1) - 8 (sin A + 1) = 0
(10 sin A - 8) (sin A + 1) = 0
sin A = 4/5 or -1
- If sin A = -1
A = -π/2
- Now we have sec A = 1/ cos A = 1/(cos(-π/2)) = 1/ 0 = undefined.
- So consider sin A = 4/5
- Then cos A = 3/5 ..................(by pythagoras theorem)
- So sin A + cos A = 4/5 + 3/5 = 7/5
Answer:
So the value of sin A + cos A is 7/5.
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