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Basic Trig FunctionsAn Introduction to Sine, Cosine, and Tangent
Melanie Forbes
Mosque Detail, provided by Let’s Book (Jamie Carter) on Flickr, Creative Commons, some rights reserved, April 2009, Seoul, Korea.
What are Trig Functions? Let’s look at 3 similar triangles
If the angles of one triangle are exactly the same as the angles of another, those triangles are similar.
3
4
5
θ
β
6
8
10
θ
β
4.5
6
7.5
θ
β
These are 3 similar right triangles; they each have a 90° angle, equal angles θ, and equal angles β.
Comparing Sides Look at the green triangle and find angle θ Compare the side opposite θ to the
hypotenuse
3
4
5
θ
β
Comparing Sides Look at the taupe triangle and find angle θ Compare the side opposite θ to the
hypotenuse
4.5
6
7.5
θ
β
Ratios are the Same Did you notice that each of the 3 similar
triangles, even though the triangles were different sizes, had the exact same ratio of sides?
Comparing Sides Look at the orange triangle and find angle θ Compare the side opposite θ to the
hypotenuse
6
8
10
θ
β
The Sine Function Ancient mathematicians noticed that
similar right triangles always had side ratios that were the same.
In a right triangle, the ratio of the side opposite a given angle to the hypotenuse is known as the Sine (abbreviated sin) ratio.
Sine θ = In this triangle sin θ = And sin β =
3
4
5
θ
β
Mosaic Tile Medallion, provided by A. Davey on flickr, Creative Commons, Some rights reserved, April 2010, Mosque, Esfahan, Iran.
Practice In each triangle, determine sine θ.
Just click to see the correct answer
What is Sine θ?
Sin θ =
7
2425
θ
What is Sine θ?
Sin θ =
9
40
41θ
What is Sine θ?
Sin θ =
5
12
13
θ
How did you do? If you correctly identified the sine ratios
in those triangles, click to move to the next section.
Otherwise, click below to go back.
Go to the beginning Review the sine ratio
Left Arrows Reverse Button, Provided by: Microsoft
Decorative and Calligraphic Tilewormk, provided by A. Davey on flickr, Creative Commons, Some rights reserved, April 2010, Mosque, Esfahan, Iran.
The Cosine Function
In a right triangle, the ratio of the side adjacent to a given angle compared to the hypotenuse is known as the Cosine (abbreviated cos) ratio.
Cosine θ = In this triangle cos θ = And cosβ = 3
4
5
θ
β
Practice In each triangle, determine cosine θ.
Just click to see the correct answer
What is Cosine θ?
Cos θ =
7
2425
θ
What is Cosine θ?
Cos θ =
9
40
41θ
What is Cosine θ?
Cos θ =
5
12
13
θ
How did you do? If you correctly identified the cosine
ratios in those triangles, click to move to the next section.
Otherwise, click below to go back.
Review the sine ratio Review the cosine ratio
Left Arrows Reverse Button, Provided by: Microsoft
3D Ornamental Mosaic Tile, provided by A. Davey on flickr, Creative Commons, Some rights reserved, April 2010, Mosque, Kerman, Iran
The Tangent Ratio
In a right triangle, the ratio of the side opposite a given angle compared to the side adjacent to the angle is known as the Tangent (abbreviated tan) ratio.
Tangent θ = In this triangle tan θ = And tanβ = 3
4
5
θ
β
Practice In each triangle, determine tangent θ.
Just click to see the correct answer
What is Tangent θ?
Tan θ =
7
2425
θ
What is Tangent θ?
Tan θ =
9
40
41θ
What is Tangent θ?
Tan θ =
5
12
13
θ
How did you do? If you correctly identified the tangent
ratios in those triangles, click to move to the next section.
Otherwise, click below to go back.
Review the cosine ratio
Review the tangent ratio
Left Arrows Reverse Button, Provided by: Microsoft
SOH – CAH - TOA You can remember the definitions of the 3
trig ratios with a mnemonic
SOH: Sin=Opposite/Hypotenuse
CAH: Cos=Adjacent/Hypotenuse
TOA: Tan=Opposite/Adjacent
Detail 3-D Tilework, provided by A. Davey on flickr, Creative Commons, Some rights reserved, April 2010, Mosque, Esfahan, Iran.
Thanks for Watching.
Any idea why I included pictures from the Middle East?
Detail 3-D Tilework, provided by A. Davey on flickr, Creative Commons, Some rights reserved, April 2010, Mosque, Esfahan, Iran.
