I need help on these problems they are from my study guide and i don't have a clue on how to do these 3 problems and if anyone can show the steps and somewhat explain the steps so i can do it myself next time.
1.) sin25�‹cos35�‹sin�‹35
The answer is �ã3/2
2.) tan10�‹+tan20�‹/1-tan10�‹tan20�‹
The answer is �ã3/3
3.) cos(7ƒÎ/12)+cos(5ƒÎ/12)+sin7ƒÎ/12)sin(5ƒÎ12)
The answer is �ã3/2
I don't know why the format is like this but the symbol behind the sin25 are degrees,the letter a on the answers are root, and the fl letter on number 3 are suppose to be PIE
1. First of all sin25cos35sin35 is NOT equal to √3/2
I know sin 60° = √3/2, so I looked at your expression to see what you might have meant.
since 25+35 = 60, you probably meant
sin25°cos35° + cos35°sin25°
which would be sin(25+35)° = sin60° = √3/2
2. again a typo, you meant
(tan10+tan20)/(1-tan10tan20))
by the tan(A+B) expansion
(tan10+tan20)/(1-tan10tan20))
= tan(10+20)°
=tan 30°
= 1/√3 , which is the same as your √3/3
3. once again a typo
Should have been
cos(7π/12)cos(5π/12) + sin(7π/12)sins5π/12)
= cos(7π/12 - 5π/12) , by the cos(A-B) expansion
= cos π/6 or cos 30°
=√3/2
Sure, let's go through each problem step by step and explain how to solve them.
1.) sin25°cos35°sin35°
To solve this problem, you can use the trigonometric identity sin(A)cos(B) = (1/2) * [sin(A-B) + sin(A+B)]. In this case, A = 25° and B = 35°.
Step 1: Substitute the values into the formula:
(1/2) * [sin(25°-35°) + sin(25°+35°)]
Step 2: Simplify:
(1/2) * [sin(-10°) + sin(60°)]
Step 3: Use the trigonometric identity sin(-θ) = -sin(θ):
(1/2) * [-sin(10°) + sin(60°)]
Step 4: Evaluate the remaining trigonometric functions:
(1/2) * [-sin(10°) + √3/2]
Step 5: Simplify and combine like terms:
(-1/2) * sin(10°) + (√3/4)
The final answer is (√3/4) - (1/2) * sin(10°), which cannot be simplified further.
2.) tan10° + tan20° / 1 - tan10°tan20°
To solve this problem, we'll use the trigonometric identity tan(A+B) = (tanA + tanB) / (1 - tanA*tanB). In this case, A = 10° and B = 20°.
Step 1: Substitute the values into the formula:
(tan(10°) + tan(20°)) / (1 - tan(10°)tan(20°))
Step 2: Evaluate the tangent functions:
(tan(10°) + tan(20°)) / (1 - tan(10°)tan(20°))
Step 3: Simplify and combine like terms:
(tan(10°) + tan(20°)) / (1 - tan(10°)tan(20°))
The final answer is (tan(10°) + tan(20°)) / (1 - tan(10°)tan(20°)), which cannot be simplified further.
3.) cos(7π/12) + cos(5π/12) + sin(7π/12)sin(5π/12)
To solve this problem, we'll use the trigonometric identity cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2). In this case, A = 7π/12 and B = 5π/12.
Step 1: Substitute the values into the formula:
2cos((7π/12 + 5π/12)/2)cos((7π/12 - 5π/12)/2) + sin(7π/12)sin(5π/12)
Step 2: Simplify the angles:
2cos(12π/24)cos(2π/24) + sin(7π/12)sin(5π/12)
Step 3: Evaluate the cosine and sine functions:
2cos(π/2)cos(π/12) + sin(7π/12)sin(5π/12)
Step 4: Simplify and combine like terms:
2 * 0.5 * √3/2 + sin(7π/12)sin(5π/12)
Step 5: Evaluate the remaining trigonometric functions:
√3/2 + sin(7π/12)sin(5π/12)
The final answer is √3/2 + sin(7π/12)sin(5π/12), which cannot be simplified further.