Given sine of alpha=2/3 and cosine of alpha is less than zero, find the exact value of the other five trigonometric functions.
sketch a right-angled triangle with hypotenuse 3 and opposite side as 2
by Pythagoras the adjacent side to angle alpha is √5
You should know the definition of the 6 trig ratios in terms of opposite, adjacent and hypotenuse
e.g. tan (alpha)= 2/√5
To find the exact value of the other five trigonometric functions (cosine, tangent, cotangent, secant, and cosecant), given the sine of alpha and that the cosine of alpha is less than zero, we can use the Pythagorean identity and the definitions of the trigonometric functions.
Given that sine of alpha (sin(alpha)) = 2/3, we can define the opposite side as 2 and the hypotenuse as 3.
Since cosine of alpha (cos(alpha)) is less than zero, it means that the adjacent side is negative. According to the Pythagorean identity, sin²(alpha) + cos²(alpha) = 1, we can find the value of the adjacent side.
sin(alpha) = opposite/hypotenuse
=> 2/3 = opposite/3
=> opposite = 2
Using the Pythagorean identity,
cos²(alpha) = 1 - sin²(alpha)
cos²(alpha) = 1 - (2/3)²
cos²(alpha) = 1 - 4/9
cos²(alpha) = 9/9 - 4/9
cos²(alpha) = 5/9
Taking the square root of both sides,
cos(alpha) = ±sqrt(5/9)
Since we know that cosine is less than zero, we take the negative value,
cos(alpha) = -sqrt(5/9)
Now, we can find the remaining trigonometric functions:
1. Tangent (tan(alpha)) = sin(alpha) / cos(alpha)
tan(alpha) = (2/3) / (-sqrt(5/9))
tan(alpha) = -2sqrt(5) / 3
2. Cotangent (cot(alpha)) = 1 / tan(alpha)
cot(alpha) = 1 / (-2sqrt(5)/3)
cot(alpha) = -3 / (2sqrt(5))
3. Secant (sec(alpha)) = 1 / cos(alpha)
sec(alpha) = 1 / (-sqrt(5/9))
sec(alpha) = -sqrt(9/5)
4. Cosecant (csc(alpha)) = 1 / sin(alpha)
csc(alpha) = 1 / (2/3)
csc(alpha) = 3/2
Therefore, the exact value of the other five trigonometric functions is:
cos(alpha) = -sqrt(5/9)
tan(alpha) = -2sqrt(5) / 3
cot(alpha) = -3 / (2sqrt(5))
sec(alpha) = -sqrt(9/5)
csc(alpha) = 3/2