Given that theta = (5)/(13), and theta is acute, find the value of ( theta + theta)/( theta).
|Mathematics
Given that theta = (5)/(13), and theta is acute, find the value of ( theta + theta)/( theta).
ScanSolve AI Answer
4 steps
Answer
60209
Here are the solutions to your trigonometry problems:
Given that sinθ=135, and θ is acute, find the value of sinθtanθ+cosθ.
Step 1: Find cosθ and tanθ.
Since θ is acute, we use the identity sin2θ+cos2θ=1.
cos2θ=1−sin2θ=1−(135)2=1−16925=169169−25=169144
Since θ is acute, cosθ>0.
cosθ=169144=1312
Now, find tanθ:
tanθ=cosθsinθ=12/135/13=125
Step 2: Substitute the values into the expression sinθtanθ+cosθ.
135125+1312
Step 3: Simplify the expression.
First, simplify the numerator:
125+1312=12×135×13+12×12=15665+144=156209
Now, divide by sinθ:
135156209=156209×513
Since 156=12×13:
12×13209×513=12×5209=6020960209
Find x if sin5x=cos4x.
Step 1: Use the complementary angle identity cosA=sin(90∘−A).
sin5x=sin(90∘−4x)
Step 2: Equate the angles.
5x=90∘−4x
Step 3: Solve for x.
5x+4x=90∘9x=90∘x=990∘x=10∘10∘
Determine the value of θ if sin(20∘+θ)=cos(30∘+θ).
Step 1: Use the complementary angle identity cosA=sin(90∘−A).
cos(30∘+θ)=sin(90∘−(30∘+θ))cos(30∘+θ)=sin(90∘−30∘−θ)cos(30∘+θ)=sin(60∘−θ)
Step 2: Equate the angles.
sin(20∘+θ)=sin(60∘−θ)20∘+θ=60∘−θ
Step 3: Solve for θ.
θ+θ=60∘−20∘2θ=40∘θ=240∘θ=20∘20∘
Given that sinθ=53, and θ is acute, find the value of secθ+cotθ.
Step 1: Find cosθ, secθ, and cotθ.
Since θ is acute, we use the identity sin2θ+cos2θ=1.
cos2θ=1−sin2θ=1−(53)2=1−259=2525−9=2516
Since θ is acute, cosθ>0.
cosθ=2516=54
Now, find secθ:
secθ=cosθ1=4/51=45
Next, find tanθ:
tanθ=cosθsinθ=4/53/5=43
Then, find cotθ:
cotθ=tanθ1=3/41=34
Step 2: Substitute the values into the expression secθ+cotθ.
45+34
Step 3: Simplify the expression.
45+34=4×35×3+4×4=1215+16=12311231
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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 your trigonometry problems: 1. Given that = (5)/(13), and is acute, find the value of ( + )/( ). Step 1: Find and . Since is acute, we use the identity ^2 + ^2 = 1. ^2 = 1 - ^2 = 1 - ((5)/(13))^2 = 1 - (25)/(169) = (169 - 25)/(169) = (144)/(169) Since is acute, > 0. = sqrt((144)/(169)) = (12)/(13) Now, find : = ( )/( ) = (5/13)/(12/13) = (5)/(12) Step 2: Substitute the values into the expression ( + )/( ). (5)/(12) + (12)/(13)(5)/(13) Step 3: Simplify the expression. First, simplify the numerator: (5)/(12) + (12)/(13) = (5 × 13 + 12 × 12)/(12 × 13) = (65 + 144)/(156) = (209)/(156) Now, divide by : (209)/(156)(5)/(13) = (209)/(156) × (13)/(5) Since 156 = 12 × 13: (209)/(12 × 13) × (13)/(5) = (209)/(12 × 5) = (209)/(60) (209)/(60) 2. Find x if 5x = 4x. Step 1: Use the complementary angle identity A = (90^ - A). 5x = (90^ - 4x) Step 2: Equate the angles. 5x = 90^ - 4x Step 3: Solve for x. 5x + 4x = 90^ 9x = 90^ x = (90^)/(9) x = 10^ 10^ 3. Determine the value of if (20^ + ) = (30^ + ). Step 1: Use the complementary angle identity A = (90^ - A). (30^ + ) = (90^ - (30^ + )) (30^ + ) = (90^ - 30^ - ) (30^ + ) = (60^ - ) Step 2: Equate the angles. (20^ + ) = (60^ - ) 20^ + = 60^ - Step 3: Solve for . + = 60^ - 20^ 2 = 40^ = (40^)/(2) = 20^ 20^ 4. Given that = (3)/(5), and is acute, find the value of + . Step 1: Find , , and . Since is acute, we use the identity ^2 + ^2 = 1. ^2 = 1 - ^2 = 1 - ((3)/(5))^2 = 1 - (9)/(25) = (25 - 9)/(25) = (16)/(25) Since is acute, > 0. = sqrt((16)/(25)) = (4)/(5) Now, find : = (1)/( ) = (1)/(4/5) = (5)/(4) Next, find : = ( )/( ) = (3/5)/(4/5) = (3)/(4) Then, find : = (1)/( ) = (1)/(3/4) = (4)/(3) Step 2: Substitute the values into the expression + . (5)/(4) + (4)/(3) Step 3: Simplify the expression. (5)/(4) + (4)/(3) = (5 × 3 + 4 × 4)/(4 × 3) = (15 + 16)/(12) = (31)/(12) (31)/(12) That's 2 down. 3 left today — send the next one.
