IF ALPHA +BETA=PI/2 THEN SIN ALPHA =1/3 THEN SIN BETA =?
since a + b = π/2
a and b are complimentary angles
thus sin b = cos a
we know sin a = 1/3
make a sketch of your right-angled triangle, find the adjacent side to be √8
thus cos a = √8/3
so sin b = √8/3 or 2√2/3
To find the value of sin(beta) based on the given information, we can use the trigonometric identity:
sin(beta) = sin(pi/2 - alpha).
Given that sin(alpha) = 1/3, we can use this value to find sin(beta).
Step 1: Substitute the given values into the trigonometric identity.
sin(beta) = sin(pi/2 - alpha)
Step 2: Convert alpha to radians.
To find alpha in radians, we can use the formula: radians = degrees * pi/180.
Since sin(alpha) = 1/3, we can find alpha using the inverse sine function (or arcsin function).
alpha = arcsin(1/3) ≈ 0.3398 radians
Step 3: Substitute the value of alpha into the trigonometric identity.
sin(beta) = sin(pi/2 - alpha)
sin(beta) = sin(pi/2 - 0.3398)
Step 4: Simplify the expression by evaluating sin(pi/2 - 0.3398).
Using the trigonometric identity sin(pi/2 - x) = cos(x):
sin(beta) = cos(0.3398)
Step 5: Calculate the value of cos(0.3398).
To find cos(0.3398), we can use a calculator or a trigonometric table.
cos(0.3398) ≈ 0.940
Therefore, sin(beta) ≈ 0.940.
To find the value of sin(beta), we can use the trigonometric identity of complementary angles: sin(beta) = sin(pi/2 - alpha).
Given that sin(alpha) = 1/3 and alpha + beta = pi/2, we need to find the value of alpha first.
sin(alpha) = 1/3
We can solve for alpha by taking the inverse sine (arcsin) of both sides:
arcsin(sin(alpha)) = arcsin(1/3)
alpha = arcsin(1/3)
Next, we substitute the value of alpha into the equation alpha + beta = pi/2:
arcsin(1/3) + beta = pi/2
Now, to find the value of beta, we can subtract arcsin(1/3) from both sides:
beta = pi/2 - arcsin(1/3)
Finally, we substitute the value of beta into the equation sin(beta) = sin(pi/2 - alpha):
sin(beta) = sin(pi/2 - arcsin(1/3))
Using the trigonometric identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b), we can simplify the equation further:
sin(beta) = sin(pi/2)cos(arcsin(1/3)) - cos(pi/2)sin(arcsin(1/3))
Since sin(pi/2) = 1 and cos(pi/2) = 0, the equation simplifies to:
sin(beta) = 1 * cos(arcsin(1/3)) - 0 * sin(arcsin(1/3))
sin(beta) = cos(arcsin(1/3))
Now, we can use the trigonometric identity cos(arcsin(x)) = sqrt(1 - x^2):
sin(beta) = sqrt(1 - (1/3)^2)
sin(beta) = sqrt(1 - 1/9)
sin(beta) = sqrt(8/9)
sin(beta) = sqrt(8) / sqrt(9)
sin(beta) = sqrt(8) / 3
Therefore, sin(beta) = sqrt(8) / 3 or approximately 0.9428.