Use the special triangles to find the exact values of the following:
1) csc45degrees
2)sec30degrees
3)longest side of a 30-60-90
triangle if the side opposite the 60degree angle is 5radical3
Looks like you have several questions of this type.
At the top of your notebook, draw two triangles
the 45-45-90 and the 30-60-90 triangle with their sides.
Without any further work you can now find the 6 trig ratios of 30° , 45°, and 60°
1) and 2) become trivial after that
e.g. csc 45° = 1/sin45° = 1/ (1/√2) = √2
3) set up a ratio between your triangle and the basic 30-60-90 triangle, that is, ....
x/2 = 5√3/√3
x/2 = 5
x = 10
Or by the "just look at it" concept, ....
notice that your side is 5 times as big as the basic side, so your longest side must be 5 times the hypotenuse of the basic triangle. 5x2 = 10
To find the exact values of trigonometric functions using special triangles, we can use two commonly known triangles: the 45-45-90 triangle and the 30-60-90 triangle.
1) csc 45 degrees:
The 45-45-90 triangle has two congruent legs and a hypotenuse. The angles in this triangle are 45 degrees, 45 degrees, and 90 degrees. The ratio of the hypotenuse to each leg is √2.
To find the csc (cosecant) of an angle, we take the reciprocal of the sin (sine) of that angle. The sin of 45 degrees is 1/√2. Hence, the csc of 45 degrees is the reciprocal of 1/√2, which simplifies to √2.
Therefore, csc 45 degrees = √2.
2) sec 30 degrees:
The 30-60-90 triangle has angles measuring 30 degrees, 60 degrees, and 90 degrees. In this triangle, the ratio of the side lengths is 1 : √3 : 2.
The ratio of the hypotenuse to the adjacent side (which corresponds to sec) is 2.
Thus, sec 30 degrees = 2.
3) Longest side of a 30-60-90 triangle:
In a 30-60-90 triangle, the side lengths are in the ratio 1 : √3 : 2.
If the side opposite the 60 degree angle is given as 5√3, we can determine the longest side of the triangle by multiplying the given side by 2.
The longest side of a 30-60-90 triangle would be 2 * 5√3 = 10√3.