1. Use special right triangles to state the value of the 6 trig functions for 30 degrees, 45 degrees and 60 degrees.
2. Use the calculator to evaluate the value of the following trig functions.
sec 54^o 23’
cot 35^o 21’ 45’’
seems pretty straightforward. Where do you get stuck?
Your trig text has all the special angles, and the rest you can do on your calculator.
The only trick is to
(a) convert ° ' " to °
35°21'45" = 35 + 21/60 + 45/3600 = 2829/80 °
(b) recall that cotx = 1/tanx
1. To use special right triangles to determine the values of the trigonometric functions, we need to consider two types of right triangles: the 45-45-90 triangle and the 30-60-90 triangle.
For a 45-45-90 triangle, the two legs are congruent, and the hypotenuse is equal to √2 times the length of either of the legs. In this triangle, the angles are 45 degrees.
For a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. In this triangle, the angles are 30 degrees, 60 degrees, and 90 degrees.
Now, let's determine the values of the trigonometric functions for the given angles:
For 30 degrees:
- Sin(30°) = (opposite side length) / (hypotenuse length) = 1/2
- Cos(30°) = (adjacent side length) / (hypotenuse length) = √3/2
- Tan(30°) = Sin(30°) / Cos(30°) = (1/2) / (√3/2) = 1/√3
- Csc(30°) = 1 / Sin(30°) = 2
- Sec(30°) = 1 / Cos(30°) = 2/√3
- Cot(30°) = 1 / Tan(30°) = √3
For 45 degrees:
Since a 45-45-90 triangle has congruent sides, the trigonometric values are the same for both 45 degrees and 90 degrees.
- Sin(45°) = (opposite side length) / (hypotenuse length) = 1/√2
- Cos(45°) = (adjacent side length) / (hypotenuse length) = 1/√2
- Tan(45°) = Sin(45°) / Cos(45°) = 1
- Csc(45°) = 1 / Sin(45°) = √2
- Sec(45°) = 1 / Cos(45°) = √2
- Cot(45°) = 1 / Tan(45°) = 1
For 60 degrees:
- Sin(60°) = (opposite side length) / (hypotenuse length) = √3/2
- Cos(60°) = (adjacent side length) / (hypotenuse length) = 1/2
- Tan(60°) = Sin(60°) / Cos(60°) = (√3/2) / (1/2) = √3
- Csc(60°) = 1 / Sin(60°) = 2/√3
- Sec(60°) = 1 / Cos(60°) = 2
- Cot(60°) = 1 / Tan(60°) = 1/√3
2. To evaluate trigonometric functions using a calculator, follow these steps:
- First, make sure your calculator is set to the correct angle mode (degrees or radians). For these examples, we will assume degrees.
- Enter the angle in the calculator using degrees, minutes, and seconds format.
- Press the appropriate function key for the trigonometric function you want to calculate (e.g., sin, cos, tan, csc, sec, cot).
- Lastly, press Enter or equals to get the result.
Let's evaluate the given trigonometric functions:
- sec 54° 23':
- Enter 54° 23' into the calculator.
- Press the sec key on the calculator.
- Press Enter or equals to get the result.
- cot 35° 21' 45'':
- Enter 35° 21' 45'' into the calculator.
- Press the cot key on the calculator.
- Press Enter or equals to get the result.