Trigonometry Chapter 7

Review of right triangle relationships  Right triangles have very specific relationships.  We have learned about the Pythagorean relationship that exists in right triangles.  There is another very useful relationship that can be used to find an unknown side of a triangle.  In mathematics it is referred to as the “tangent ratio”

7.2 The Tangent Ratio  The relationship between the angle in a right triangle and the lengths of the sides of the triangle.  The angle and two sides must be in a very specific pattern for the ratio to be meaningful.  Tangent θ = opposite side / adjacent side  The 90 degree angle is never the angle used in the calculation.

New way of looking at a right triangle  Hypotenuse is still always the longest side of the triangle.  Opposite and adjacent depend on which angle you are looking from,

Tangent from different angles  Let look at triangle ABC If you are looking from angle A then line BC is opposite from it. This means that line AB has to be the adjacent side since hypotenuse has to be the longest side. If you are looking from angle C then line AB is opposite from it. This makes line AB the opposite side since hypotenuse has to be the longest side in the right triangle. **This means you have to pay attention to which angle you are using.

Finding a side length using the tangent ratio. Tan (angle) = opposite / adjacent To find the side length you need to know the angle and one of the sides. **Do not use hypotenuse. It is not part of the tangent ratio. Each triangle is a bit different and you need to take a moment with each triangle to make sure you are setting the ratio up correctly.

Example 1:  Find the length of side x.

Example 2:  Find the length of side x

Example 3  Find the measure of side X if angle A is 46 degrees.

Practice Problems  Page 340 # 1-9

Finding an angle using tangent ratio  If you are given a right triangle and the opposite and adjacent sides are given you can find the missing angle by using the reverse of the tangent ratio  To do this we use the tan -1 button on the calculator.  Press “shift” and then the tan button on the calculator.  The value you get is the angle instead of the length of a side.  It is actually less work than finding a side length!

Example 1:  Find the measure of the missing angle indicated by “x” Use θ = tan -1 (opp/adj) X = tan -1 (12.6/9.5) X = tan -1 (1.326) X = 53⁰

Example 2:  Find the missing angle in the following triangle. Use θ = tan -1 (opp/adj) X = tan -1 (4.5/11.8) X = tan -1 (0.381) X = 20.9⁰

7.3 Sine and Cosine Ratios  Trigonometry also makes use of other ratios to find missing sides and angles of a triangle.  By using these ratios very few measurements are needed to be able to find all angles and sides in a triangle.  These relationships are similar to tangent but the sides involved are different.

All Right Triangle Relationships  With these three relationships you can always find all sides and angles of a right triangle Most people remember these using SOH CAH TOA **In this course these relationships will be give during assessment

Lets try some examples  Can you find the length of the missing side of this triangle? First, label the sides of the triangle as opposite, adjacent, and hypotenuse. (Make sure you do this from looking at the angle.) This time we have opposite and hypotenuse sides we need to work with. SOH CAH TOA When you have opposite and hypotenuse we need to use sine!

Solving for a side using sine ratio sin θ = (opp/hyp) sin 47 = (28/x) X sin 47 = 28 X = 28/sin 47 X = 38.3

Solving for a side using cosine ratio cos θ = (adj/hyp) cos 32 = (X/72.5) 72.5 (cos 32) = X X = 61.5

Finding a missing angle using sine ratio Use θ = sin -1 (opp/hyp) X = sin -1 (48/65.4) X = sin -1 (0.734) X = 47.2⁰

Finding a missing angle using cosine ratio Use θ = cos -1 (adj/hyp) X = cos -1 (12.8/21.7) X = cos -1 (0.590) X = 53.9⁰

Practice using sine, cosine, and tangent  Worksheet on trigonometry ratios (posted on blog)  Worksheet on inverse trigonometry ratios (posted on blog)