1. Lesson 4.2
There are TWO historical perspectives of trigonometry which incorporate different methods for introducing trigonometric functions. Our first introduction is based on the unit circle…

2. Warm Up 4.2 What is the Pythagorean Theorem?
What is the purpose of the Pythagorean Theorem?
Draw the triangles and Solve for the missing triangle lengths: ) ) 10 5 2 12 6 3

3. With your Partner… use the Pythagorean theorem to solve for the missing sides as indicated

4. Recall from geometry: What is the measure of each interior angle of an equilateral triangle?
Draw an equilateral triangle with 1 inch side lengths and label the sides and angles with the appropriate measures
60°
60°
60°

5. Draw an altitude that cuts the triangle in half
Draw an altitude that cuts the triangle in half. What are the interior angles and side measurements of the halved triangle?
Use the Pythagorean Theorem to solve for this missing side
60°
30°
1 in
1 in
1 in
60°
60°
60°
90°
1 in
1/2 in

6. Recall from geometry: What is the measure of each interior angle of a square?
Draw a square with side lengths that measure inches and label the sides and angles with the appropriate measures
90°
90°
90°
90°

7. What happens when we cut the square in half with a diagonal?
Use the Pythagorean Theorem to solve for this missing side
90°
90°
45°
1 𝑖𝑛 in in
90°
90°
90°
45° in in

8. Finding Special Values
Answers on textbook pg #270
Sin(θ)
cos(θ)
Use your triangle templates to find all sin and cos values of the special angles
In other words…you will find and label the coordinates of the points that correspond to the intersections of the special angles and the unit circle. See the example .

9. A circle given by the formula x²+ y² = 1 (radius is 1 unit)
Appearing or occurring at intervals
The length of one complete cycle

10. A point (x, y) on the unit circle
(1, 0)
Central angle
The real number “t” is the length of the arc intercepted
by the angle Ɵ, given in radians.

11. Fun Fact: Where did the
word “sine” come from?

12. Sin (x) – refers to the height of a point on a circle at a particular level of inclination
Cos (x) – refers to the “overness” of a point on a circle at a particular level of inclination
“jya”
“jiba”
“sine”
Scholars of ancient India wanted to know how far away the sun was. They discovered that if you have an object moving in a circular arc we can solve for heights.
10th century math took off in the middle east and they translated what the Indian scholars had done, translated this word for “height” into their language, “jiba.”
Then the latin people made a mistake in translating, because they though the word “jiba” meant harbor in Arabic. So they used the latin word for harbor, which is “sine“
Wonderful story of mistranlation!

13. sine
tangent
cosine

14. Complete both tables with your group members and your unit circle!

15. To help you remember which quadrant has positive sin/cos/tan values

18. What values of x and y always stay between?
Domain – because you can substitute any θ into these functions (you can go around the unit circle an infinite amount of times) , the domain of the sin(θ) and cos(θ) functions is the set of all real numbers.
Range – because the radius of the unit circle is 1, the values of sin(θ) and cos(θ) will always range between -1 and 1.
This shows the sin(θ) will always be between -1 and 1
This shows the cos(θ) will always be between -1 and 1

19. All real numbers
-1 ≤ y ≤ 1
All real numbers
-1 ≤ y ≤ 1 2П 2П

20. Use 1/cos(x) because sec(x) is the reciprocal of cos(x).
To evaluate csc(x), sec(x), and cot(x) on the calculator, use
1/[the reciprocal function of sin(x), cos(x), or tan(x)]

21. Practice!
Half slip of paper

22. Add in 2π to the end of your charts What is the value?
Because the values begin to repeat, we
say the periods of the sine and cosine
functions are 2π

23. A circle given by the formula x²+ y² = 1 (radius is 1 unit)
Appearing or occurring at intervals
The length of one complete cycle

25. Using specific points to explore the period of tangent
Add a row to your table as shown
Fill in those values
Mark the values of tangent
which are the same.