if 13 cos a=12, find the the value of sec a+sina?
cosa = 12/13
construct a right-angled triangle and use Pythagoras to find the sides 5, 12, and 13
since the cosine is + in I or IV, we have two cases:
case1: in quad I
sina = 5/13, cosa = 12/13
seca + sina
= 13/12 + 5/13
= 229/156
case2: in quad IV
sina = -5/13, cosa = 12/13
seca + sina
= 13/12 - 5/13
= 109/156
Let , 13 cos A = 12
Cos A = adjacent/ hypo
= 12/13
AB = 12 , AC = 13
By using Pythagoras theorem
AC² = AB² + BC²
(13)² = (12)² + BC²
169 = 144 + BC²
169 - 144 = BC²
25 = BC²
BC = 5
(i) Sin A = opposite / hypo
= BC/AC
★ CONTINUE ★
= 5/13
(ii) Sec A = hypo / adj
= AC/ AB
= 13/12
I think that case1 is the one, even though I am not sure math can be sometimes hard
Case1
To find the value of sec(a) + sin(a) given that 13cos(a) = 12, we can use trigonometric identity and substitution.
1. Start with the given equation: 13cos(a) = 12
2. Divide both sides of the equation by 13 to isolate cos(a): cos(a) = 12/13
3. Recall the trigonometric identity: sec^2(a) = 1 + tan^2(a)
4. Substitute the value of cos(a) into the identity: sec^2(a) = 1 + [tan^2(a)] = 1 + [sin^2(a)/cos^2(a)]
5. Take the square root of both sides of the equation to get sec(a): sec(a) = sqrt(1 + sin^2(a)/cos^2(a))
6. Substitute the value of cos(a) from step 2 into the equation: sec(a) = sqrt(1 + sin^2(a)/(12/13)^2)
7. Simplify the equation: sec(a) = sqrt(1 + sin^2(a)/144/169)
8. Multiply the numerator and denominator of the fraction by 169 to clear the fraction: sec(a) = sqrt((169 + 169*sin^2(a))/144)
9. Take the square root of both sides of the equation to get sin(a): sqrt((169 + 169*sin^2(a))/144) = sec(a)
10. Square both sides of the equation to eliminate the square root: (169 + 169*sin^2(a))/144 = sec^2(a)
11. Remember the identity: sec^2(a) = 1 + tan^2(a), and substitute it into the equation: (169 + 169*sin^2(a))/144 = 1 + tan^2(a)
12. Rearrange the equation to isolate tan^2(a): (169 + 169*sin^2(a))/144 - 1 = tan^2(a)
13. Simplify the equation: (169*sin^2(a) + 169 - 144)/144 = tan^2(a)
14. Combine like terms: (169*sin^2(a) + 25)/144 = tan^2(a)
15. Substitute the value of cos(a) from step 2 into the equation: (169*sin^2(a) + 25)/144 = 1 - cos^2(a)
16. Simplify the equation: (169*sin^2(a) + 25)/144 = 1 - (12/13)^2
17. Solve for sin^2(a): 169*sin^2(a) + 25 = 144 - 144*(12/13)^2
18. Simplify the equation: 169*sin^2(a) + 25 = 144 - 144*(144/169)
19. Combine like terms: 169*sin^2(a) + 25 = 144 - 144*144/169
20. Perform the multiplications: 169*sin^2(a) + 25 = 144 - 144*144/169
21. Simplify further: 169*sin^2(a) + 25 = 144 - 20736/169
22. Combine the terms with a common denominator: 169*sin^2(a) + 25 = (144*169 - 20736)/169
23. Distribute and combine on the right side: 169*sin^2(a) + 25 = (24336 - 20736)/169
24. Simplify on the right side: 169*sin^2(a) + 25 = 3600/169
25. Multiply both sides by 169: 169*(169*sin^2(a) + 25) = 3600
26. Distribute on the left side: 169*sin^2(a) + 25*169 = 3600
27. Simplify: 169*sin^2(a) + 4225 = 3600
28. Subtract 4225 from both sides: 169*sin^2(a) = 3600 - 4225
29. Combine like terms: 169*sin^2(a) = -625
30. Divide by 169 to isolate sin^2(a): sin^2(a) = -625/169
31. Since sin^2(a) cannot be negative, there is no real solution for sin(a).
Therefore, there is no real solution for sec(a) + sin(a) when 13cos(a) = 12.