If 5cosA+3=0 and 180⁰<A<306⁰ determine by means of a diagram tan²A
You can calculate tan² A
5 cos A + 3 = 0
Subtract 3 to both sides
5 cos A = - 3
Divide both sides by 5
cos A = - 3 / 5
For angles 180⁰ < A < 360⁰
The cosine is negative in quadrant III and positive in quadrant IV.
Angle A lies in quadrant III ( 180⁰ < A < 270⁰ )
cos² A = 9 / 25
Use basic trigonometric identity:
sin² A + cos² A =1
sin² A + 9 / 25 = 1
Subtract 9 / 25 to both sides
sin² A = 1 - 9 / 25
sin² A = 25 / 25 - 9 / 25
sin² A = 16 / 25
tan² A = sin² A / cos² A
tan² A = ( 16 / 25 ) / ( 9 / 25 )
tan² A = 16 / 9
5cosA+3=0 , where A is in quadrant III or IV
cosA = -3/5 , but the cosine is negative in III, so A must be in III
now construct a diagram in quadrant III
with x = -3, y = ? and r = 5
Use Pythagoras or you should recognize the 3-4-5 right-angled triangle, so
sinA = -4/5
tan A = sinA/cosA = (-4/5) / (-3/5) = 4/3
tan^2 A = 16/9
To determine the value of tan²A using the given equation, we first need to solve for A. Let's rewrite the equation and solve for cosA:
5cosA + 3 = 0
Subtracting 3 from both sides,
5cosA = -3
Dividing both sides by 5,
cosA = -3/5
Since cosine is negative, we know that A is in either the second or third quadrant.
To find the value of A, we can use the inverse cosine function (cos⁻¹) to find the angle where cosine is equal to -3/5. Using a calculator, cos⁻¹(-3/5) ≈ 131.8°.
Since we know that 180° < A < 306°, we can deduce that A is in the third quadrant.
Now that we know the value of A, we can find tanA by using the tangent function (tanA = sinA/cosA). We already have cosA, so let's find sinA.
To find sinA, we can use the Pythagorean identity: sin²A + cos²A = 1. Rearranging the equation, sin²A = 1 - cos²A.
sin²A = 1 - (-3/5)²
sin²A = 1 - 9/25
sin²A = 25/25 - 9/25
sin²A = 16/25
Taking the square root of both sides,
sinA = ±√(16/25)
sinA = ±4/5
Since A is in the third quadrant, sinA is negative.
sinA = -4/5
Now, we can find tanA:
tanA = sinA/cosA
tanA = (-4/5) / (-3/5)
tanA = 4/3
Finally, to find tan²A:
tan²A = (tanA)²
tan²A = (4/3)²
tan²A = 16/9
Therefore, tan²A = 16/9.
To determine the value of tan²A, we need to find the value of tan A first.
Given that the equation 5cosA + 3 = 0, we can start by isolating the cosine term. Subtracting 3 from both sides of the equation gives us:
5cosA = -3
Next, we divide both sides of the equation by 5 to solve for cos A:
cosA = -3/5
To find the value of tan A, we can use the identity: tan A = sin A / cos A.
However, we first need to find the value of sin A in order to calculate tan A. We can use the Pythagorean identity: sin²A + cos²A = 1.
Since we already know the value of cos A (cosA = -3/5), we can substitute this value into the identity:
sin²A + (-3/5)² = 1
sin²A + 9/25 = 1
Next, isolate sin²A by subtracting 9/25 from both sides:
sin²A = 1 - 9/25
sin²A = 25/25 - 9/25
sin²A = 16/25
Taking the square root of both sides, we find:
sin A = √(16/25)
sin A = 4/5
Now that we have the values of cos A and sin A, we can calculate tan A:
tan A = sin A / cos A
tan A = (4/5) / (-3/5)
tan A = 4/5 * -5/3
tan A = -4/3
Finally, to find the value of tan²A, we square the value of tan A:
tan²A = (-4/3)²
tan²A = 16/9
Therefore, tan²A is equal to 16/9.
To summarize the steps:
1. Start with the equation 5cosA + 3 = 0 and isolate the cosine term.
2. Solve for cos A by dividing both sides by 5.
3. Use the Pythagorean identity sin²A + cos²A = 1 to find sin A.
4. Substitute the value of cos A into the identity and solve for sin A.
5. Calculate tan A using the formula tan A = sin A / cos A.
6. Square the value of tan A to find tan²A.