Right Angle Trigonometry

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  Hypotenuse Adjacent Opposite  The _________ is always the longest side of the triangle.  The _________ side is the leg directly across from the angle.  The _________ side is the leg alongside the angle. hypotenuse opposite adjacent

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

SohCahToa S ine equals O pposite over H ypotenuse C osine equals A djacent over H ypotenuse T angent equals O pposite over A djacent

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

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

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 trig ratios

Steps to finding the missing sides of a right triangle using trigonometric ratios: 1. Redraw the figure and mark on it HYP, OPP, ADJ relative to the given angle 61  9.6 cm x HYP OPP ADJ

Steps to finding the missing sides of a right triangle using trigonometric ratios: 2. For the given angle choose the correct trigonometric ratio which can be used to set up an equation 3. Set up the equation 61  9.6 cm x HYP OPP ADJ

Steps to finding the missing sides of a right triangle using trigonometric ratios: 4. Solve the equation to find the unknown. 61  9.6 cm x HYP OPP ADJ

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

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

Steps to finding the missing angle of a right triangle using trigonometric ratios: 1. Redraw the figure and mark on it HYP, OPP, ADJ relative to the unknown angle  5.92 km HYP OPP ADJ 2.67 km

Steps to finding the missing angle of a right triangle using trigonometric ratios: 2. For the unknown angle choose the correct trig ratio which can be used to set up an equation 3. Set up the equation  5.92 km HYP OPP ADJ 2.67 km

Steps to finding the missing angle of a right triangle using trigonometric ratios: 4. Solve the equation to find the unknown using the inverse of trigonometric ratio.  5.92 km HYP OPP ADJ 2.67 km

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

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

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

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

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

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

Practice: Other Figures (Trapezoid)  Find x given: 10 cm x cm 65  48 

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