Given that cos a=5/13 and that 270<a>360 evaluate tan a +sin a
To evaluate tan a + sin a, we need to first find the value of sin a.
From the given information, we have cos a = 5/13. Using the Pythagorean identity sin^2 a + cos^2 a = 1, we can calculate sin a as follows:
sin^2 a + cos^2 a = 1
sin^2 a + (5/13)^2 = 1
sin^2 a + 25/169 = 1
sin^2 a = 1 - 25/169
sin^2 a = 144/169
sin a = ± √(144/169)
sin a = ± 12/13
Since a is in the range of 270° to 360°, the sine function is negative. Therefore, sin a = -12/13.
Now we can calculate tan a + sin a:
tan a + sin a = sin a/cos a + sin a
= (-12/13) / (5/13) + (-12/13)
= -12/5 - 12/13
= (-12*13 - 12*5) / (5*13)
= (-156 - 60) / 65
= -216 / 65
So, tan a + sin a = -216/65.
Given that tan theatre=12/5 and theater is a reflex,find the value of 5 sin thetre + cos thetre
If tan θ = 12/5 and θ is a reflex angle, we know that θ is in the third quadrant. In the third quadrant, both sin θ and cos θ are negative.
We are asked to evaluate 5sinθ + cosθ.
First, let's find the values of sin θ and cos θ. Since tan θ = 12/5, we can use the Pythagorean identity to find sin θ and cos θ:
tan θ = sin θ / cos θ
12/5 = sin θ / cos θ
Cross multiplying, we get:
5sin θ = 12cos θ
Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, we substitute 5sin θ with 12cos θ:
25sin^2 θ + cos^2 θ = 25(12cos θ)^2 + cos^2 θ = 144cos^2 θ + cos^2 θ
= 145cos^2 θ
Now, we have:
145cos^2 θ = 1 - sin^2 θ
145cos^2 θ = 1 - (1 - cos^2 θ)
145cos^2 θ = cos^2 θ
144cos^2 θ = 0
cos θ = 0
Since θ is in the third quadrant, cos θ is negative. Therefore, cos θ = -1.
Now, we can find sin θ:
sin^2 θ + cos^2 θ = 1
sin^2 θ + (-1)^2 = 1
sin^2 θ + 1 = 1
sin^2 θ = 0
sin θ = 0
We have found that cos θ = -1 and sin θ = 0.
Now, we can evaluate 5sin θ + cos θ:
5sin θ + cos θ = 5(0) + (-1)
= -1
Therefore, 5sin θ + cos θ = -1.
Solve for x in the equation:tan x=2 sin x for -180<x>180
To solve for x in the equation tan x = 2 sin x, we can rewrite it using the identity sin x / cos x = 2 sin x:
sin x / cos x = 2 sin x
Now, we can cross multiply:
sin x = 2 sin x * cos x
Dividing both sides by sin x, we get:
1 = 2 cos x
Dividing both sides by 2, we have:
1/2 = cos x
Since the range of x is -180° to 180°, we know that in the second and third quadrants, the cosine function is negative. Therefore, we have:
cos x = -1/2
From the unit circle, we know that the angle whose cosine is -1/2 is 120° or 2π/3 radians. However, we need to take into account the range given: -180° to 180°.
In the second quadrant, the reference angle is 60° or π/3 radians, so one solution is:
x = 180° - 60° = 120°
In the third quadrant, the reference angle is also 60° or π/3 radians, so another solution is:
x = -180° - 60° = -240°
Therefore, the solutions for x in the equation tan x = 2 sin x are x = 120° and x = -240°.
To evaluate tan a + sin a, we need to find the values of tan a and sin a.
Given that cos a = 5/13, we can use the Pythagorean identity to calculate sin a:
sin a = √(1 - cos^2 a)
= √(1 - (5/13)^2)
= √(1 - 25/169)
= √(144/169)
= 12/13
Next, we can use the identity tan a = sin a / cos a to find tan a:
tan a = sin a / cos a
= (12/13) / (5/13)
= (12/13) * (13/5)
= 12/5
Finally, we can substitute the values into the expression tan a + sin a:
tan a + sin a = (12/5) + (12/13)
To get a common denominator, we can multiply the first term by 13/13 and the second term by 5/5:
= (12/5) * (13/13) + (12/13) * (5/5)
= (156/65) + (60/65)
= 216/65
Therefore, tan a + sin a = 216/65.
To evaluate tan(a) + sin(a), we need to find the values of the tangent and sine of angle a.
We are given that cos(a) = 5/13. Since cos(a) is positive and cos and sec are positive in the fourth quadrant between 270° and 360°, we can conclude that a lies in the fourth quadrant.
To find the value of sin(a), we can use the Pythagorean identity sin^2(a) + cos^2(a) = 1.
Since we know cos(a) = 5/13, we can solve this equation for sin(a):
sin^2(a) + (5/13)^2 = 1
sin^2(a) + 25/169 = 1
sin^2(a) = 1 - 25/169
sin^2(a) = 144/169
sin(a) = ± √(144/169)
sin(a) = ± (12/13)
Since a lies in the fourth quadrant, sin(a) will be negative. Therefore, sin(a) = -12/13.
Now, let's evaluate tan(a):
tan(a) = sin(a) / cos(a)
tan(a) = (-12/13) / (5/13)
tan(a) = (-12/13) * (13/5)
tan(a) = -12/5
Finally, we can calculate the desired expression:
tan(a) + sin(a) = -12/5 + (-12/13)
tan(a) + sin(a) = (-12/5) * (13/13) + (-12/13) * (5/5)
tan(a) + sin(a) = (-156/65) + (-60/65)
tan(a) + sin(a) = -216/65
Therefore, tan(a) + sin(a) = -216/65.