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# Trigonometric Ratios

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### Trigonometric Ratios

1. 1. Trigonometric Ratios Trigonometry – Mrs. Turner Hipparcus – 190 BC to 120 BC – born in Nicaea (now Turkey) was a Greek astronomer who is considered to be one of the first to use trigonometry.
2. 2. Parts of a Right Triangle Hypotenuse side Opposite side A B C Now, imagine that you move from angle A to angle B still facing into the triangle. Imagine that you, the happy face, are standing at angle A facing into the triangle. The hypotenuse is neither opposite or adjacent . You would be facing the opposite side and standing next to the adjacent side. You would be facing the opposite side and standing next to the adjacent side. Opposite side Adjacent side Yasuaki Japanese Mathematician
3. 3. Review Hypotenuse Hypotenuse Opposite Side Adjacent Side A B For Angle A This is the Opposite Side This is the Adjacent Side For Angle B A This is the Adjacent Side This is the Opposite Side Opposite Side Adjacent Side B Hilda Hudson British Mathematician
4. 4. Trig Ratios We can use the lengths of the sides of a right triangle to form ratios. There are 6 different ratios that we can make. Using Angle A to name the sides Use Angle B to name the sides The ratios are still the same as before!! A B Hypotenuse Adjacent Side opposite Szasz Hungarian Mathematician
5. 5. Trig Ratios <ul><li>Each of the 6 ratios has a name </li></ul><ul><li>The names also refer to an angle </li></ul>Hypotenuse Adjacent Opposite A Birkhoff American Mathematician Sine of Angle A = Cosine of Angle A = Tangent of Angle A = Cosecant of Angle A = Secant of Angle A = Cotangent of Angle A =
6. 6. Trig Ratios Hypotenuse Adjacent Opposite If the angle changes from A to B The way the ratios are made is the same B Freitag German Mathematician Sine of Angle = Cosine of Angle = Tangent of Angle = Cosecant of Angle = Secant of Angle = Cotangent of Angle = B B B B B B
7. 7. Trig Ratios <ul><li>Sine, Cosine and Tangent ratios are the most common. </li></ul>Adjacent Opposite A Hypotenuse <ul><li>Each of these ratios has an abbreviation </li></ul>Sin A = Cos A = Tan A = Csc A= Sec A = Cot A = Sine of Angle A = Cosine of Angle A = Tangent of Angle A = Cosecant of Angle A = Secant of Angle A = Cotangent of Angle A = John Dee English Mathematician
8. 8. SOHCAHTOA Adjacent A B Opposite Hypotenuse Here is a way to remember how to make the 3 basic Trig Ratios 1) Identify the Opposite and Adjacent sides for the appropriate angle <ul><li>SOHCAHTOA is pronounced “Sew Caw Toe A” and it means </li></ul><ul><li>S in is O pposite over H ypotenuse, C os is A djacent over H ypotenuse, and T an is O pposite over A djacent </li></ul>Use the underlined letters to make the word SOH-CAH-TOA Quetelet Flemish Mathematician
9. 9. Examples of Trig Ratios 6 10 8 A B First we will find the Sine, Cosine and Tangent ratios for Angle A. Next we will find the Sine, Cosine, and Tangent ratios for Angle B Adjacent Opposite Remember SohCahToa Lame French Mathematician Sin A = Cos A = Tan A = Sin B = Cos B = Tan B =
10. 10. Examples of Trig Ratios 10 8 A B Now, we will find the Cosecant, Secant and Cotangent ratios for Angle A. Next we will find the Cosecant, Secant, and Cotangent ratios for Angle B Adjacent Opposite Remember SohCahToa backwards 6 Benneker African American Mathematician Csc B = Sec B = Cot B = Csc A = Sec A = Cot A =
11. 11. Special Triangles A B The short side is always opposite the smaller angle. In this triangle, angle B is smaller than angle A Hypotenuse Hypotenuse A B In this triangle, Angle A is smaller than angle B Albertus German Mathematician
12. 12. Special Triangles There are two special triangles: The 30-60-90 triangle and the 45-45-90 triangle. These two right triangles are used often, so you should memorize the lengths of the sides opposite these angles. Alberti Italian Mathematician
13. 13. 30-60-90 If one of the acute angles is 30 ̊, the other must be 60 ̊. When the side opposite the 30 ̊ angle is 1 unit, then the side opposite the 60 ̊ angle is units and the hypotenuse is 2 units. 1 2 Nasir Islamic Mathematician
14. 14. 30-60-90 First, we will write the trigonometric ratios of the angle that measures 30 ̊. Second, we will write the trigonometric ratios of the angle that measures 60 ̊. Remember Soh-Cah-Toa Oleinik Ukraine Mathematician 1 2 Sin 30 ̊ = Cos 30 ̊ = Tan 30 ̊ = Sin 60 ̊ = Cos 60 ̊ = Tan 60 ̊ =
15. 15. 45-45-90 If one acute angle of a right triangle is 45 ̊, then the other acute angle must be 45 ̊̊. 1 1 Cristoffel French Mathematician If the side opposite one 45 ̊ is 1 unit, then the side opposite the other 45 ̊ is also 1 unit. The hypotenuse is
16. 16. 45-45-90 First, we will write the trigonometric ratios of the angle that measures 45 ̊. Since the other acute angle is also 45 ̊, the ratios will be the same. Battaglini Italian Mathematician Sin 45 ̊ = Cos 45 ̊ = Tan 45 ̊ = 1 1