1. Elevation and depression

2. Agenda
Introduction to trigonometry- Right-angled triangles, theta, etc.
Trigonometric functions
Angle of elevation
Angle of depression
Applicability in real life
Simple problems involving angles of elevation/depression

3. Introduction to Trigonometry
Formed from Greek words 'trigonon' (triangle) and 'metron' (measure). 
Trigonometric triangles are always right-angled triangles

4. More on Trigonometry A branch of mathematics that studies
Triangles
Relationship between sides and angles between sides
A branch of mathematics that studies
Describes relationship between sides/angles
Uses trigonometric functions

5. Sides of a Right-angled Triangle
Opposite to the right-angle
Longest side
Hypotenuse
Side that touches θ
Adjacent
Side opposite to θ
Opposite

6. Theta 8th letter of the Greek alphabet Represented by “θ”
A variable, not a constant
Commonly used in trigonometry to represent angle values

7. Trigonometric Functions
Sin (Sine)= ratio of opposite side to the hypotenuse
Cos (Cosine)= ratio of adjacent side to the hypotenuse
Tan (Tangent)= ratio of opposite side to the adjacent side

8. Easier way to remember Sin, Tan, Cos
SOH CAH TOA
TOA: Tangent =  Opposite ÷ Adjacent (T=O/A)
CAH: Cosine = Adjacent ÷ Hypotenuse (C=A/H)
SOH: Sine = Opposite ÷ Hypotenuse (S=O/H)

9. Angle of Elevation
The angle of elevation is the angle between the horizontal line and the observer’s line of sight, where the object is above the observer

10. Angle of Depression
The angle of depression is the angle between the horizontal line and the observer’s line of sight, where the object is below the observer

11. Applicability of Angles of Elevation and Depression
Used by architects to design buildings by setting dimensions
Used by astronomers for locating apparent positions of celestial objects
Used in computer graphics by designing 3D effects properly
Used in nautical navigations by sailors (sextants)
Many other uses in our daily lives

12. To find sides
If given two sides of the right triangle, you can use the Pythagorean Theorem
𝑎 2 + 𝑏 2 = 𝑐 2
If given one side and one angle, you will use a trigonometric function.

13. Finding Angles
To find angles, you have to use the inverse of the trigonometric ratios.
Sin => sin −1
Cos => cos −1
Tan => tan −1

14. Example 1
Question: You see a huge tree that is 50 feet in height and it casts a shadow of length 60 feet. You are standing at the tip of the shadow. What is the degree of elevation from the end of the shadow to the top of the tree?

15. Example 2
From a point 115 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 64.3 degrees . Find the height of the tree to the nearest foot.

16. Example 3
From a point 10 feet from the base of a flag pole, the angle of elevation to the top of the flag pole is 67.4 degrees . Find the height of the flag pole to the nearest foot.

17. Example 4
If the distance from a helicopter to a tower is 300 feet and the angle of depression is 40.2 degrees , find the distance on the ground from a point directly below the helicopter to the tower

18. Example 5
The angle of depression of one side of a lake, measured from a balloon 2500 feet above the lake is 43 degrees . The angle of depression to the opposite side of the lake is 27 degrees . Find the width of the lake

19. Overall summary Draw the diagram Identify the known values
Form equations
Solve

20. We hope you have enjoyed our presentation
Thank you for your kind attention! Please ask reasonable questions, if any.