1. Unit 1B Right Triangle Trigonometry & Bearings

2. Agenda (Teacher: Modify Due/Link for Extra Credit) Todays Date:
Warm-up (10 minutes)
Learning Targets (2 minutes)
Right Triangle Trigonometry Discussion & Practice (30 minutes)
Bearings Discussion & Practice (15 minutes)
Quizizz Knowledge Check: Right Triangle Trig and Bearings
Homework: MathXL Unit 1B Day 1 Right Triangle Trig
Extra Credit Opportunity: PSAT Practice Problems
Due:
Link:

3. Warm-up Please take out a sheet of paper.
Over the next 5 minutes, you are going back to your Geometry days…
Take two minutes & write down everything you remember from geometry about right triangle trigonometry.
Pick a partner and for 1 minute one of you share your remembrances and then switch and have the other partner add on to your list for the next 30 seconds.
Once you are completed, find the perfect squares below without using your calculator :
4 = = = = = = =
225 = = = = = = =
25 = = = = = =
After 5 minutes we will spend another 2-3 minutes sharing as a class.

4. Learning Targets: Right Triangle Trigonometry and Bearings
Simplify and reduce radicals and rationalize the denominators of a fraction
Find the 6 basic trigonometric ratios using the given information
Find all angles/legs of a right triangle using the given information and the 6 trigonometric ratios
Be able to draw and label the reference triangle and its reference angle from the given information
Given Information may include: standard position angle, reference angle, reference point, side or angle of a triangle, the angle of elevation or depression or a trigonometric ratio.
Understand the difference between the standard angle measure and a bearing and learn how to interpret and write a bearing from it’s north line only or it’s north-south line.

5. 1st Concept: Simplifying & Rationalizing Radicals
Many answers in trigonometry will contain radicals ( ). These radicals must be simplified and if there is a radical in the denominator it must be rationalized.
Simplifying Radicals:
Leave in radical form (no rounded decimal answers from your calculator)
Pull out perfect squares
Examples:
4 = = =
You Practice:
25 = = =

6. 1st Concept (cont’d): Simplifying & Rationalizing Radicals
B) Rationalizing Radicals:
Leave no radical in the denominator of a fraction
Multiple the top & bottom of fraction by the simplified radical in the denominator
Eliminate complex fractions
Simplify to the lowest term
Examples:
= = = =
You Practice:
= = = =

7. 2nd Concept: Angle of Elevation & Depression
Angle of Elevation: The angle above the horizontal that an observer must look up to, to see an object. Angle of Depression: The angle below the horizontal that an observer must look down to see an object.
Illustration:

8. 3rd Concept: Trigonometric Ratios: SOH-CAH-TOA
sin 𝜃 = 𝑂 𝐻 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒
cos 𝜃 = 𝐴 𝐻 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒
tan 𝜃 = 𝑂 𝐴 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Helpful Saying:
csc 𝜃 = 𝐻 𝑂 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
s𝑒𝑐 𝜃 = 𝐻 𝐴 = 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑒𝑢𝑠𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cot 𝜃 = 𝐴 𝑂 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
Helpful Saying:

9. Example 1: Given all sides of a triangle, find the values of all 6 trigonometric functions of angle Ɵ

10. Example 2: Given that cot θ = 5 12 , evaluate the remaining five trig functions of angle Ɵ.

11. Example 3: Given a = 5 and C = 82°, find the values of each missing piece of the triangle:

12. Example 4: Lucy is flying over Round Rock High School to scout out where to build building 3500 . Her plane has a 47° angle of depression, and her elevation is 1350 feet. What is her horizontal distance to RRHS?

13. 4th Concept: Standard Position Angle, Reference Angles & Triangles
A standard position angle is formed from its initial side (or ray), which is always the positive x axis, and its terminal side (or ray). The terminal side is drawn counter clockwise from the initial side for positive angles. A reference triangle is formed by "dropping" a perpendicular line from the terminal ray of a standard position angle to the x- axis. Remember, it must be drawn to the x-axis. The reference angle is the angle inside the reference triangle closest to the origin.

14. Based on the definition we just discussed, which is the reference triangle below A or B?

15. Reference Triangle Example: Suppose θ is in standard position and (-3,7) is a point on the terminal side of θ. Draw a reference triangle, find each of the missing angles and use the reference angle to determine each of the 6 trig functions.

16. 5th Concept: Special Right Triangles
Example 1: Find the exact value of: a) sin 30° b) cos 45° You try: c) cos 60° d) tan 60°

17. Example 2: Using the concept of special right triangles find the acute angle that satisfies the given equation.
a) cos 𝜃= b) cot 𝜃=1
You try:
c) sin 𝜃= d) 𝑡𝑎𝑛𝜃=

18. Bearings:
Bearings are used to represent the angle of an object’s line of travel.
Bearings are also referred to as an object’s “course” or “direction”.
Bearings are used for measuring navigation angles for ships or planes or in orienteering.
Bearings are measured differently from standard angle measures.
Standard Angle Measure: Bearing Angle Measure (Compass):

19. North Line vs. North-South Line
There are 2 different ways to describe a bearing.
North Line Only
Acute & Obtuse Angles
North-South Line
Acute angles only
North or South first
The acute angle second
East or West last
Example: In the picture to the right:
The bearing of P from O is _______ or ____________
north line north-south line
The bearing of Q from O is _______ or ____________

20. Bearing Example: You Do !
058°
In the picture:
The bearing of A to B is _______ or ____________ Extension: What is the Standard Angle: __________
north line north-south line What is the Reference angle: __________
The bearing of B to A is _______ or ____________ Extension: What is the Standard Angle: __________
north line north-south line What is the Reference Angle: __________