Right Angled Trigonometry

1. Right Angled Trigonometry

2. Labeling a Right Triangle • In trigonometry, we give each side a name according to its position in relation to any given angle in the triangle: Hypotenuse, Opposite, Adjacent • The _________ is always the longest side of the triangle. hypotenuse Hypotenuse Adjacent • The _________ side is the leg directly across from the angle. opposite  Opposite • The _________ side is the leg alongside the angle. adjacent

3. Trigonometric Ratios We define the 3 trigonometric ratios in terms of fractions of sides of right angled triangles.  Hypotenuse (HYP) Adjacent (ADJ) Opposite (OPP)

4. SohCahToa Sine equals Opposite over Hypotenuse Cosine equals Adjacent over Hypotenuse Tangent equals Opposite over Adjacent

5. Practice Together: a 32 x Given each triangle, write the ratio that could be used to find x by connecting the angle and sides given. b x 65

6. YOU DO: d c x Given the triangle, write all the ratios that could be used to find x by connecting the angle and sides given. 56

7. In a right triangle, if we are given another angle and a side we can find: • The third angle of the right triangle: • How? • The other sides of the right triangle: • How? Using the ‘angle sum of a triangle is 180’ Using the trigonometric ratios

8. Steps to finding the missing sides of a right triangle using trigonometric ratios: • Redraw the figure and mark on it HYP, OPP, ADJ relative to the given angle 9.6 cm HYP x ADJ 61 OPP

9. Steps to finding the missing sides of a right triangle using trigonometric ratios: • For the given angle choose the correct trigonometric ratio which can be used to set up an equation • Set up the equation 9.6 cm HYP x ADJ 61 OPP

10. Steps to finding the missing sides of a right triangle using trigonometric ratios: • Solve the equation to find the unknown. 9.6 cm HYP x ADJ 61 OPP

11. Practice Together: x m Find, to 2 decimal places, the unknown length in the triangle. 7.8 m 41

12. a m YOU DO: b m 63 14.6 m Find, to 1 decimal place, all the unknown angles and sides in the triangle. 

13. Steps to finding the missing angle of a right triangle using trigonometric ratios: • Redraw the figure and mark on it HYP, OPP, ADJ relative to the unknown angle 5.92 km OPP ADJ 2.67 km HYP 

14. Steps to finding the missing angle of a right triangle using trigonometric ratios: • For the unknown angle choose the correct trig ratio which can be used to set up an equation • Set up the equation 5.92 km OPP ADJ 2.67 km HYP 

15. Steps to finding the missing angle of a right triangle using trigonometric ratios: • Solve the equation to find the unknown using the inverse of trigonometric ratio. 5.92 km OPP ADJ 2.67 km HYP 

16. Practice Together: 3.1 km Find, to one decimal place, the unknown angle in the triangle. 2.1 km 

17. YOU DO: 7 m 4 m Find, to 1 decimal place, the unknown angle in the given triangle. 

18. Practice: Isosceles Triangles • Using what we already know about right angles in isosceles triangles find the unknown side. x cm 67 10 cm

19. YOU DO: Isosceles Triangles • Find the unknown angle of the isosceles triangle using what you already know about right angles in isosceles triangles. 5.2 m 8.3 m 

20. Practice: Circle Problems • Use what you already know about right angles in circle problems to find the unknown angle. 10 cm 6 cm 

21. YOU DO: Circle Problems • Use what you already know about right angles in circle problems to find the unknown side length. 6.5 cm 56 x cm

22. Practice: Other Figures (Trapezoid) • Find x given: x cm 10 cm 65 48

23. YOU DO: Other Figures (Rhombus) • A rhombus has diagonals of length 10 cm and 6 cm respectively. Find the smaller angle of the rhombus.  6 cm 10 cm