Solving Right Triangles using 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 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 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 trigonometric 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 by cross multiplying. 61  9.6 cm x HYP OPP ADJ *Remember you answer had to be less than 9.6cm

Practice Together: Find, to 2 decimal places, the unknown length in the triangle. 41  x m 7.8 m 1. Since the given sides are opposite and adjacent, use tangent. 2. Cross multiply to solve for x.

YOU DO: Find, to 1 decimal place, all the unknown angles and sides in the triangle.  a m 14.6 m 63  b m 1. First the angles sum to 180° so Θ = Use tangent to find b. 3. Use sine to find a.

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 opposite adjacent *on a graphing calc hit 2 nd tan, then 2.1 divided by 3.1 *some other calculators require you to hit 2.1 divided by 3.1 then shift tan.

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

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  1. I know that a rhombus has diagonals that bisect each other and are perpendicular 2. I then have a right triangle with opposite side 3 and adjacent side 5 3. This means I will use inverse tangent to solve for the angle.

Summary  If you are solving for a missing side, you set up your trig ratio and cross multiply  If x is in the numerator, then you multiply  If x is in the denominator, then you divide  If you are solving for a missing angle, you set up your trig ratio and use the inverse trig key  If x is inside the triangle, use the inverse key