Evaluate tan^2 60 degrees +sin^2 45 degrees without using your calculator.
a)5
b)7/3
c)1/2
d)7/2
Please help.
Draw the 30-60 and 45-45 triangles. What is the answer? You ought to be able to do these from memory. If you cant, make and use flash cards on these triangles.
To evaluate tan^2 60 degrees + sin^2 45 degrees without using a calculator, we need to use the trigonometric identities.
First, let's draw the 30-60-90 triangle and the 45-45-90 triangle:
/|
/ |
/ |
√3 / |
/ |
/ |
/ |
C /______| B
√3 A
/|
/ |
/ |
√2 / |
/ |
/ |
/45°|
C /______|B
√2 A
In the 30-60-90 triangle, side lengths are in the ratio 1:√3:2. Since angle C is 60 degrees, the opposite side (the side opposite to angle C) would have length √3. Therefore, side C is √3, side B is 1, and side A is 2.
In the 45-45-90 triangle, side lengths are in the ratio 1:1:√2. Since angle C is 45 degrees, both sides C and B would have length 1. Therefore, side A is √2.
Now let's evaluate the expressions:
tan^2 60 degrees:
First, let's find the value of tan 60 degrees. In the 30-60-90 triangle, tan 60 degrees is equal to opposite side (side C, which is √3) divided by the adjacent side (side B, which is 1). Therefore, tan 60 degrees = √3/1 = √3.
Hence, tan^2 60 degrees = (√3)^2 = 3.
sin^2 45 degrees:
In the 45-45-90 triangle, sin 45 degrees is equal to the ratio of the opposite side (side C, which is 1) to the hypotenuse (side A, which is √2). Therefore, sin 45 degrees = 1/√2.
To simplify sin^2 45 degrees, we square the value of sin 45 degrees:
sin^2 45 degrees = (1/√2)^2 = 1/2.
Now, let's compute the expression:
tan^2 60 degrees + sin^2 45 degrees = 3 + 1/2 = 6/2 + 1/2 = 7/2.
Therefore, the correct answer is option (d) 7/2.