Evaluate the trigonometric functions by memory or by contruscting appropriate triangles for the given special angles.
(a) csc 30deg
(b) sin pi/4
Somewhere in your notes or your textbook you should have diagrams of the 30-60-90 and the 45-45-90 right-angled triangles and their sides.
remember csc30° = 1/sin30° , which you should know
and π/4 =45°, and you should know that as well
To evaluate trigonometric functions, such as csc and sin, for special angles, it helps to memorize the values of these functions for commonly used angles. For example, the values of sin, cos, tan, sec, csc, and cot for 0°, 30°, 45°, 60°, and 90° are usually memorized.
(a) To evaluate csc 30°, we can construct an equilateral triangle and find the vertical side length opposite the 30° angle.
In an equilateral triangle, all sides are equal, and each angle measures 60°. Therefore, the length of any side is the same as the length of the vertical line opposite a 30° angle.
If we take an equilateral triangle and draw an altitude from one of the vertices to the midpoint of the opposite side, it divides the equilateral triangle into two 30°-60°-90° triangles.
For this triangle, we can use the relationships among the sides: the length of the hypotenuse is twice the length of the shorter side, and the length of the longer side is equal to the length of the shorter side multiplied by √3.
In this triangle, the length of the shorter side opposite the 30° angle is 1, so the length of the hypotenuse is 2, and the length of the longer side is 1 multiplied by √3, which is √3.
Since csc is the reciprocal of sin, we know that csc 30° is equal to 1/sin 30°. By memorizing the values of sine for special angles, we know that sin 30° is equal to 1/2.
Therefore, csc 30° = 1/(1/2) = 2.
(b) To evaluate sin (π/4), we can construct a right triangle with an angle of π/4 radians.
In a right triangle with an angle of π/4 radians (45°), the two legs are congruent, and the hypotenuse is the square root of 2 times the length of one of the legs. The lengths of the sides can be set according to our convenience.
Let's assume that one of the legs is 1. Then, the hypotenuse can be determined using the Pythagorean theorem:
(1)^2 + (1)^2 = c^2,
Simplifying, we get:
1 + 1 = c^2,
2 = c^2,
c = √2.
Therefore, the sine of π/4 radians is equal to the length of the opposite side divided by the hypotenuse, which is 1/√2.
To simplify this, we multiply both the numerator and denominator by √2:
(1/√2) * (√2/√2) = √2/2.
Hence, sin (π/4) = √2/2.