How do I get exact measures?
cosine 270 degrees
sin 90 degrees
tan 360 degrees
csc 45 degrees
no calculator!
The best way to solve these problems is to draw a unit circle (radius = 1), and calculate these ratios from the definitions (cosine = adjacent / hypotenuse, etc.)
The first three can be solved easily this way. The last one,
csc(45°)=1/sin(45°)
Knowing that sin(45)=cos(45), and
sin²(45)+cos²(45)=1, we solve for sin(45)=(√2)/2, so
csc(45)=2/√2 = √2.
To get the exact measures of trigonometric functions without using a calculator, you will need to use the unit circle and the special angles.
First, let's define each of the trigonometric functions:
- Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
- Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
- Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
- Cosecant (csc): The cosecant of an angle is the reciprocal of the sine of that angle.
Now, let's find the exact measures of the provided angles without using a calculator:
1. Cosine 270 degrees:
- 270 degrees is in the third quadrant of the unit circle.
- In the unit circle, the cosine is negative in the third quadrant.
- The exact value of cosine 270 degrees is -1.
2. Sine 90 degrees:
- 90 degrees is in the second quadrant of the unit circle.
- In the unit circle, the sine is positive in the second quadrant.
- The exact value of sine 90 degrees is 1.
3. Tangent 360 degrees:
- 360 degrees is a full circle rotation, which brings us back to the same position on the unit circle.
- The tangent of a full circle rotation is 0.
- The exact value of tangent 360 degrees is 0.
4. Cosecant 45 degrees:
- Cosecant is the reciprocal of sine.
- To find csc 45 degrees, we need to find the reciprocal of the sine of 45 degrees.
- The sine of 45 degrees is (√2)/2 (which is obtained from the special right triangle).
- The reciprocal of (√2)/2 is 2/(√2), which simplifies to √2.
Therefore, the exact measures without using a calculator are:
- cos 270 degrees = -1
- sin 90 degrees = 1
- tan 360 degrees = 0
- csc 45 degrees = √2