Find the exact value of each expression(no calculator):
1) sin^2(30degrees) + 1/sec^2(20degrees)
2) sec(50degrees)*cos(50degrees)
To find the exact value of each expression without using a calculator, we can utilize trigonometric identities and special triangles.
1) sin^2(30 degrees) + 1/sec^2(20 degrees):
First, let's determine the values of sin(30 degrees) and sec(20 degrees).
Using the special right triangle, an equilateral triangle cut in half, we know that the side opposite a 30-degree angle is half the length of the hypotenuse. Therefore, sin(30 degrees) = 1/2.
Next, we need to evaluate sec(20 degrees). Since sec(theta) is the reciprocal of cos(theta), we need to find cos(20 degrees).
To determine the value of cos(20 degrees), we can use the angle addition identity: cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
Let's rewrite cos(20 degrees) as cos(30 degrees - 10 degrees):
cos(30 degrees - 10 degrees) = cos(30 degrees)cos(10 degrees) - sin(30 degrees)sin(10 degrees).
Using the values we already know, cos(30 degrees) = sqrt(3)/2 and sin(30 degrees) = 1/2.
cos(30 degrees)cos(10 degrees) - sin(30 degrees)sin(10 degrees) = (sqrt(3)/2)(cos(10 degrees)) - (1/2)(sin(10 degrees)).
To evaluate further, we need to determine the exact value of cos(10 degrees). Unfortunately, this cannot be easily derived using special triangles, so we will leave it in terms of cos(10 degrees) for now.
Substituting all these values back into the expression, we have:
sin^2(30 degrees) + 1/sec^2(20 degrees) = (1/2)^2 + 1/(cos(10 degrees))^2 = 1/4 + 1/(cos(10 degrees))^2.
Therefore, the exact value of the expression is 1/4 + 1/(cos(10 degrees))^2.
2) sec(50 degrees) * cos(50 degrees):
Using similar reasoning, let's find the value of sec(50 degrees) and cos(50 degrees).
First, we need to determine cos(50 degrees). Again, we will use the angle addition identity:
cos(50 degrees) = cos(30 degrees + 20 degrees) = cos(30 degrees)cos(20 degrees) - sin(30 degrees)sin(20 degrees).
We already know that cos(30 degrees) = sqrt(3)/2 and sin(30 degrees) = 1/2. To evaluate further, we need to determine sin(20 degrees) and cos(20 degrees).
Using the right triangle, we can see that the side opposite a 30-degree angle is half the hypotenuse. Similarly, we can say that the side opposite a 60-degree angle is sqrt(3)/2 times the hypotenuse, so the side opposite a 20-degree angle would be half of that: (sqrt(3)/2)/2 = sqrt(3)/4.
Therefore, sin(20 degrees) = sqrt(3)/4.
Next, we need to find cos(20 degrees). Since cos(theta) = sin(90 degrees - theta), we can say that cos(20 degrees) = sin(90 degrees - 20 degrees) = sin(70 degrees).
Using the same reasoning as before, the side opposite a 70-degree angle is sqrt(3)/2 times the hypotenuse. Therefore, sin(70 degrees) = sqrt(3)/2.
Let's substitute the known values back into the expression:
sec(50 degrees) * cos(50 degrees) = (1/cos(50 degrees)) * cos(50 degrees) = 1.
Therefore, the exact value of the expression is 1.