Given (A) = 5/13 and (B) = 3/5, where A and B are acute angles, find (A+B). Given (A) = 1/3 and (B) = 1/4, find (A+B).
|Mathematics
Given (A) = 5/13 and (B) = 3/5, where A and B are acute angles, find (A+B). Given (A) = 1/3 and (B) = 1/4, find (A+B).
ScanSolve AI Answer
Answer
6563
Here are the solutions to the problems.
1. Given sin(A)=5/13 and cos(B)=3/5, where A and B are acute angles, find sin(A+B).
Step 1: Find cos(A).
Since A is an acute angle, cos(A)>0.
Using the identity sin2(A)+cos2(A)=1:
cos2(A)=1−sin2(A)cos2(A)=1−(135)2cos2(A)=1−16925cos2(A)=169169−25cos2(A)=169144cos(A)=169144=1312
Step 2: Find sin(B).
Since B is an acute angle, sin(B)>0.
Using the identity sin2(B)+cos2(B)=1:
sin2(B)=1−cos2(B)sin2(B)=1−(53)2sin2(B)=1−259sin2(B)=2525−9sin2(B)=2516sin(B)=2516=54
Step 3: Use the sum formula for sine.
The formula for sin(A+B) is sin(A+B)=sin(A)cos(B)+cos(A)sin(B).
sin(A+B)=(135)(53)+(1312)(54)sin(A+B)=6515+6548sin(A+B)=6515+48sin(A+B)=6563
The value of sin(A+B) is 6563.
2. Given tan(A)=1/3 and tan(B)=1/4, find tan(A+B).
Step 1: Use the sum formula for tangent.
The formula for tan(A+B) is tan(A+B)=1−tan(A)tan(B)tan(A)+tan(B).
tan(A+B)=1−(31)(41)31+41
Step 2: Simplify the expression.
tan(A+B)=1−121124+123tan(A+B)=1212−121127tan(A+B)=1211127tan(A+B)=127×1112tan(A+B)=117
The value of tan(A+B) is 117.
3. Solve the equation sin2(x)+cos2(x)=1 for all x.
Step 1: Recognize the fundamental trigonometric identity.
The equation sin2(x)+cos2(x)=1 is the Pythagorean identity.
Step 2: Determine the values of x for which the identity holds.
This identity is true for all real numbers x.
The solution for x is allrealnumbers.
4. Prove the identity cos(2x)=cos2(x)−sin2(x).
Step 1: Start with the angle sum formula for cosine.
The formula for cos(A+B) is cos(A+B)=cos(A)cos(B)−sin(A)sin(B).
Step 2: Substitute A=x and B=x into the formula.
cos(x+x)=cos(x)cos(x)−sin(x)sin(x)cos(2x)=cos2(x)−sin2(x)
The identity is proven.
5. Prove that tan(A−B)=1+tan(A)tan(B)tan(A)−tan(B).
Step 1: Express tan(A−B) in terms of sine and cosine.
tan(A−B)=cos(A−B)sin(A−B)
Step 2: Use the angle difference formulas for sine and cosine.
sin(A−B)=sin(A)cos(B)−cos(A)sin(B)cos(A−B)=cos(A)cos(B)+sin(A)sin(B)
Substitute these into the expression for tan(A−B):
tan(A−B)=cos(A)cos(B)+sin(A)sin(B)sin(A)cos(B)−cos(A)sin(B)
Step 3: Divide the numerator and denominator by cos(A)cos(B).
tan(A−B)=cos(A)cos(B)cos(A)cos(B)+cos(A)cos(B)sin(A)sin(B)cos(A)cos(B)sin(A)cos(B)−cos(A)cos(B)cos(A)sin(B)
Step 4: Simplify the terms.
Recall that cos(x)sin(x)=tan(x).
tan(A−B)=1+cos(A)sin(A)cos(B)sin(B)cos(A)sin(A)−cos(B)sin(B)tan(A−B)=1+tan(A)tan(B)tan(A)−tan(B)
The identity is proven.
6. Given sin(θ)=4/5 and θ is in the first quadrant, find sin(2θ) and cos(2θ).
Step 1: Find cos(θ).
Since θ is in the first quadrant, cos(θ)>0.
Using the identity sin2(θ)+cos2(θ)=1:
cos2(θ)=1−sin2(θ)cos2(θ)=1−(54)2cos2(θ)=1−2516cos2(θ)=2525−16cos2(θ)=259cos(θ)=259=53
Step 2: Find sin(2θ).
Use the double angle formula for sine: sin(2θ)=2sin(θ)cos(θ).
sin(2θ)=2(54)(53)sin(2θ)=2(2512)sin(2θ)=2524
Step 3: Find cos(2θ).
Use the double angle formula for cosine: cos(2θ)=cos2(θ)−sin2(θ).
cos(2θ)=(53)2−(54)2
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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the problems. 1. Given (A) = 5/13 and (B) = 3/5, where A and B are acute angles, find (A+B). Step 1: Find (A). Since A is an acute angle, (A) > 0. Using the identity ^2(A) + ^2(A) = 1: ^2(A) = 1 - ^2(A) ^2(A) = 1 - ((5)/(13))^2 ^2(A) = 1 - (25)/(169) ^2(A) = (169 - 25)/(169) ^2(A) = (144)/(169) (A) = sqrt((144)/(169)) = (12)/(13) Step 2: Find (B). Since B is an acute angle, (B) > 0. Using the identity ^2(B) + ^2(B) = 1: ^2(B) = 1 - ^2(B) ^2(B) = 1 - ((3)/(5))^2 ^2(B) = 1 - (9)/(25) ^2(B) = (25 - 9)/(25) ^2(B) = (16)/(25) (B) = sqrt((16)/(25)) = (4)/(5) Step 3: Use the sum formula for sine. The formula for (A+B) is (A+B) = (A)(B) + (A)(B). (A+B) = ((5)/(13))((3)/(5)) + ((12)/(13))((4)/(5)) (A+B) = (15)/(65) + (48)/(65) (A+B) = (15 + 48)/(65) (A+B) = (63)/(65) The value of (A+B) is (63)/(65). 2. Given (A) = 1/3 and (B) = 1/4, find (A+B). Step 1: Use the sum formula for tangent. The formula for (A+B) is (A+B) = ((A) + (B))/(1 - (A)(B)). (A+B) = (1)/(3) + (1)/(4)1 - ((1)/(3))((1)/(4)) Step 2: Simplify the expression. (A+B) = (4)/(12) + (3)/(12)1 - (1)/(12) (A+B) = (7)/(12)(12)/(12) - (1)/(12) (A+B) = (7)/(12)(11)/(12) (A+B) = (7)/(12) × (12)/(11) (A+B) = (7)/(11) The value of (A+B) is (7)/(11). 3. Solve the equation ^2(x) + ^2(x) = 1 for all x. Step 1: Recognize the fundamental trigonometric identity. The equation ^2(x) + ^2(x) = 1 is the Pythagorean identity. Step 2: Determine the values of x for which the identity holds. This identity is true for all real numbers x. The solution for x is all real numbers. 4. Prove the identity (2x) = ^2(x) - ^2(x). Step 1: Start with the angle sum formula for cosine. The formula for (A+B) is (A+B) = (A)(B) - (A)(B). Step 2: Substitute A=x and B=x into the formula. (x+x) = (x)(x) - (x)(x) (2x) = ^2(x) - ^2(x) The identity is proven. 5. Prove that (A-B) = ((A) - (B))/(1 + (A)(B)). Step 1: Express (A-B) in terms of sine and cosine. (A-B) = ((A-B))/((A-B)) Step 2: Use the angle difference formulas for sine and cosine. (A-B) = (A)(B) - (A)(B) (A-B) = (A)(B) + (A)(B) Substitute these into the expression for (A-B): (A-B) = ((A)(B) - (A)(B))/((A)(B) + (A)(B)) Step 3: Divide the numerator and denominator by (A)(B). (A-B) = ((A)(B))/((A)(B)) - ((A)(B))/((A)(B))((A)(B))/((A)(B)) + ((A)(B))/((A)(B)) Step 4: Simplify the terms. Recall that ((x))/((x)) = (x). (A-B) = ((A))/((A)) - ((B))/((B))1 + ((A))/((A))((B))/((B)) (A-B) = ((A) - (B))/(1 + (A)(B)) The identity is proven. 6. Given () = 4/5 and is in the first quadrant, find (2) and (2). Step 1: Find (). Since is in the first quadrant, () > 0. Using the identity ^2() + ^2() = 1: ^2() = 1 - ^2() ^2() = 1 - ((4)/(5))^2 ^2() = 1 - (16)/(25) ^2() = (25 - 16)/(25) ^2() = (9)/(25) () = sqrt((9)/(25)) = (3)/(5) Step 2: Find (2). Use the double angle formula for sine: (2) = 2()(). (2) = 2((4)/(5))((3)/(5)) (2) = 2((12)/(25)) (2) = (24)/(25) Step 3: Find (2). Use the double angle formula for cosine: (2) = ^2() - ^2(). (2) = ((3)/(5))^2 - ((4)/(5))^2 (2) = 9
