i)The angle á lies between 0 and 90 and is such that 2tan sqaure á + sec square á=5-4tan á. show that
3tan square á +4taná-4=0
and hence find the exact value of taná
ii)It is give that the angle â is such that cot(á+â) = 6.Without using a calculator, find the exact value of cot â
i couldnot sole (ii)
2tan^2(x) + sec^2(x) = 5 - 4tan(x)
2tan^2(x) + 1+tan^2(x) = 5 - 4tan(x)
3tan^2(x) + 4tan(x) - 4 = 0
(tanx+2)(3tanx-2) = 0
Now you know tan(x)
cot(a+b) = 6
(cota*cotb-1)/(cota+cotb) = 6
cota = -2, so
(-2cotb-1)/(-2+cotb) = 6
-12cotb-6 = -2+cotb
-4 = 13cotb
cotb = -4/13
or, use cota = 2/3
To solve part i) of the problem, we are given the equation:
2tan^2(á) + sec^2(á) = 5 - 4tan(á)
We can rewrite sec^2(á) as 1 + tan^2(á):
2tan^2(á) + 1 + tan^2(á) = 5 - 4tan(á)
Simplifying further:
3tan^2(á) + 1 = 5 - 4tan(á)
Rearranging the terms:
3tan^2(á) + 4tan(á) - 4 = 0
This is a quadratic equation in terms of tan(á). To solve it, we can use the quadratic formula:
tan(á) = [-b ± √(b^2 - 4ac)] / 2a
In this case, a = 3, b = 4, and c = -4. Substituting these values into the quadratic formula:
tan(á) = [-4 ± √(4^2 - 4(3)(-4))] / (2)(3)
tan(á) = [-4 ± √(16 + 48)] / 6
tan(á) = [-4 ± √64] / 6
tan(á) = [-4 ± 8] / 6
There are two possible solutions for tan(á):
1) tan(á) = (-4 + 8) / 6 = 4/6 = 2/3
2) tan(á) = (-4 - 8) / 6 = -12/6 = -2
Therefore, the exact values of tan(á) are 2/3 and -2.
For part ii) of the problem, we are given that cot(á+â) = 6. We want to find the exact value of cot(â).
Using the cotangent addition identity:
cot(á+â) = cot(á)cot(â) - 1
Substituting the given value, we have:
6 = cot(á)cot(â) - 1
Rearranging the terms:
cot(á)cot(â) = 7
We know that cot(á) = 1/tan(á). So, substituting this value in the equation, we get:
(1/tan(á)) cot(â) = 7
Multiply both sides by tan(á):
cot(â) = 7tan(á)
Since we already found the exact values for tan(á) in part i) as 2/3 and -2, we can substitute these values:
1) cot(â) = 7(2/3) = 14/3
2) cot(â) = 7(-2) = -14
Therefore, the exact value of cot(â) is 14/3 and -14.
To solve this problem, we are given the equation 2tan^2(á) + sec^2(á) = 5 - 4tan(á), and we need to show that it leads to the equation 3tan^2(á) + 3tan(á) - 4 = 0 and find the exact value of tan(á).
Let's start by manipulating the given equation. We know that sec^2(á) = 1 + tan^2(á), so we can substitute this into the equation:
2tan^2(á) + 1 + tan^2(á) = 5 - 4tan(á)
3tan^2(á) + 1 = 5 - 4tan(á)
Rearranging this equation, we get:
3tan^2(á) + 4tan(á) - 4 = 0
This matches the equation we need to prove.
Now, we can find the exact value of tan(á) by solving this quadratic equation. We can factorize it or use the quadratic formula. For simplicity, let's use the quadratic formula:
tan(á) = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 3, b = 4, and c = -4. Plugging in these values, we get:
tan(á) = (-4 ± √(4^2 - 4*3*(-4))) / (2*3)
= (-4 ± √(16 + 48)) / 6
= (-4 ± √(64)) / 6
= (-4 ± 8) / 6
Simplifying further, we get:
tan(á) = 4/6 or -12/6
Reducing the fractions, we have:
tan(á) = 2/3 or -2
Therefore, the exact value of tan(á) is 2/3 or -2.
Now, moving on to the second part of the question:
ii) We are given that cot(á+â) = 6, and we need to find the exact value of cot(â).
Since cot(θ) is the reciprocal of tan(θ), we can rewrite the equation as:
tan(á+â) = 1/6
Using the angle addition formula for tangent, we have:
tan(á+â) = (tan(á) + tan(â))/(1 - tan(á)tan(â)) = 1/6
Now, we need to find the exact value of cot(â), which is the reciprocal of tan(â). Let's rewrite the equation to solve for tan(â):
1/6 = (tan(á) + tan(â))/(1 - tan(á)tan(â))
Multiplying both sides by (1 - tan(á)tan(â)), we get:
(1 - tan(á)tan(â))/6 = tan(á) + tan(â)
Rearranging the terms, we have:
(1 - tan(á)tan(â))/6 - tan(á) = tan(â)
To simplify further, let's find a common denominator:
(1 - tan(á)tan(â))/6 - (6tan(á))/6 = tan(â)
Combining the fractions, we get:
(1 - tan(á)tan(â) - 6tan(á))/6 = tan(â)
Now, we can replace tan(á) with the given value, tan(á) = 2/3:
(1 - (2/3)tan(â) - 6(2/3))/6 = tan(â)
(1 - (2/3)tan(â) - 12/3)/6 = tan(â)
(1 - (2tan(â))/3 - 4)/6 = tan(â)
(1 - 2tan(â) - 12)/18 = tan(â)
(-11 - 2tan(â))/18 = tan(â)
Simplifying this equation, we have:
-11 - 2tan(â) = 18tan(â)
18tan(â) + 2tan(â) = -11
20tan(â) = -11
Finally, we can solve for tan(â):
tan(â) = -11/20
Therefore, the exact value of cot(â) is the reciprocal of tan(â), which is -20/11.