If sinA=3/5 and cosB=5/13 and if A and B are measures of two angles in Quadrant I, find the exact value of the following functions.
cotB =
sin2A=
3)cos(5pi/6 + B) =
tan(A - pi/4) =
draw your triangles. These are two of the simplest standard integer-sided ones. The have sides
3-4-5 and 5-12-13
So, if
sinA = 3/5, cosA = 4/5
cosB = 5/13, sinB = 12/13
Now just use your definitions and formulas:
cotB = cosB/sinB = 5/12
sin2A = 2sinAcosA = 2(3/5)(4/5) = 24/25
cos(5π/6+B) = cos(5π/6)cosB - sin(5π/6)sinB
= (-√3/2)(5/13)-(1/2)(12/13)
= -(12+5√3)/338
tan(A-π/4) = (tanA-tan π/4)/(1+tanA tan π/4)
= (3/4 - 1)/(1 + (3/4)(1))
= (1/4)/(7/4)
= 1/7
the last is -1/7
But I'm sure you saw my typo ...
To find the exact value of cotB, we can use the fact that cotB = 1/tanB.
Given that sinA = 3/5 and cosB = 5/13, we can use the Pythagorean identity sin^2A + cos^2A = 1 to find the missing values.
sinA = 3/5 implies cosA = √(1 - sin^2A) = √(1 - (3/5)^2) = 4/5.
Since A is in Quadrant I, both sinA and cosA are positive.
Now, tanA = sinA/cosA = (3/5) / (4/5) = 3/4.
Since A and B are in Quadrant I, tanB will also be positive.
Using cosB = 5/13, we can find sinB using sin^2B + cos^2B = 1.
sinB = √(1 - cos^2B) = √(1 - (5/13)^2) = 12/13.
Now, tanB = sinB/cosB = (12/13) / (5/13) = 12/5.
Therefore, cotB = 1/tanB = 1 / (12/5) = 5/12.
Now let's find sin2A.
sin2A = 2sinAcosA
= 2(3/5)(4/5)
= (2 * 3 * 4) / (5 * 5)
= 24/25.
Next, let's find cos(5π/6 + B).
Using the angle addition formula, cos(θ + φ) = cosθcosφ - sinθsinφ, we have:
cos(5π/6 + B) = cos(5π/6)cosB - sin(5π/6)sinB
= (-√3/2)(5/13) - (1/2)(12/13)
= (-√3 * 5 - 1 * 12) / (2 * 13)
= (-5√3 - 12) / 26.
Finally, let's find tan(A - π/4).
Using the angle subtraction formula, tan(θ - φ) = (tanθ - tanφ) / (1 + tanθtanφ), we have:
tan(A - π/4) = (tanA - tan(π/4)) / (1 + tanA*tan(π/4))
= (3/4 - 1) / (1 + (3/4)(1))
= (3/4 - 1) / (1 + 3/4)
= (-1/4) / (7/4)
= -1/7.
Therefore, the exact values of the given functions are:
cotB = 5/12,
sin2A = 24/25,
cos(5π/6 + B) = (-5√3 - 12) / 26,
tan(A - π/4) = -1/7.
To find the exact value of the functions, we will need to use some trigonometric identities and properties. Let's solve each problem step by step:
1) To find cotB, we need to use the definition of cotangent, which is the reciprocal of the tangent function. We know sinA = 3/5, and recall that sinA is equal to the opposite side divided by the hypotenuse in a right triangle with angle A. By using Pythagorean theorem, we can determine that the adjacent side is 4 (since 3^2 + 4^2 = 5^2). Now, we can find the tangent of A using tangent = opposite/adjacent: tanA = 3/4. Since cotangent is the reciprocal of tangent, cotB = 1/tanB = 1/(5/12) = 12/5.
2) To find sin2A, we need to use the double-angle formula for sine. The formula states that sin(2A) = 2sin(A)cos(A). Given sinA = 3/5, we can find cos(A) using the Pythagorean identity: cos(A) = sqrt(1 - sin^2(A)) = sqrt(1 - (3/5)^2) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5. Now we can substitute these values into the double-angle formula: sin(2A) = 2(3/5)(4/5) = 24/25.
3) To find cos(5pi/6 + B), we need to use the sum of angle formula for cosine. The formula states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B). In this case, A = 5pi/6 and cosB = 5/13. Since both A and B are in the first quadrant, cos(A) and sin(A) are both positive. We know that sin(A) = 3/5, therefore, cos(A) = sqrt(1 - sin^2(A)) = sqrt(1 - (3/5)^2) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5. Now we can substitute these values into the sum of angle formula: cos(5pi/6 + B) = (4/5)(5/13) - (3/5)(sinB).
4) To find tan(A - pi/4), we need to use the difference of angle formula for tangent. The formula states that tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)). We know that tanA = 3/4 and A is in the first quadrant, so tan(A) is positive. Now we can substitute the given values into the formula: tan(A - pi/4) = (3/4 - tan(pi/4)) / (1 + (3/4)(1)).
By following these steps, you should be able to find the exact values of the given functions.