By Mr.Bullie

Trigonometry Trigonometry describes the relationship between the side lengths and the angle measures of a right triangle. Right triangles are triangles with one 90° angle The longest side, “c,” of a right triangle is the hypotenuse. The other two sides, “a and b,” of a right triangle are the legs.

Trigonometric Ratios Trig ratios (sine, cosine, and tangent) are functions that represent the relationship between the side lengths and the angle measures θ is the Greek symbol named theta that is used to represent an unknown angle measure sin θ = opposite side / hypotenuse cos θ = adjacent side / hypotenuse tan θ = opposite side / adjacent side sin A = a/c cos A = b/c tan A = a/b sin B = b/c cos B = a/c tan B = b/a

Calculating Trig Ratios When given the measure of an angle in a right triangle, we can use trig functions to find the ratio of one side length to another. Example – to find the sin of 30°, on your calculator type “sin,” then “30,” and “exe.” So, sin 30° = 0.5

Finding Unknown Angle Measures Using Trig Because we’re given all three side lengths, we can use any trig function. So, we’ll use sine. On our calculator we hit “shift”  “sin”  “(75/85)”  “EXE” If we use the other functions, it will still produce the same answer. Try it… Example 1 – Find the measure of the indicated angle Solution – Sin θ = 75/85 θ = 61.9°

Finding Unknown Angle Measures Using Trig (Cont’d) Since we don’t have all three side lengths, we have to make sure that we choose the correct trig function to use. We are given the length of the hypotenuse and the length of the side that is adjacent to θ. So, we must use cosine. Now, on our calculator we hit “shift”  “cos”  “(12/25)”  “EXE” Example 2 – Find the measure of the indicated angle. Solution – Cos θ = 12/25 θ = 61.3°

You Try 1 2

Finding Unknown Side Lengths of a Right Triangle Using Trig Since x is the length of the side that is adjacent to the 34° angle and 18 is the length of the hypotenuse, then we must use the cosine function to solve for x. Now, we must use algebra to solve for x. Example 1 – Find the value of x. Solution – cos 34° = x / (cos 34°) = x 14.9 = x

Finding Unknown Side Lengths of a Right Triangle Using Trig (Cont’d) Since 12 is the length of the side opposite the 24° angle and x is the length of the hypotenuse, then we must use the sine function to solve for x. So, we’ll say sin 24° = 12/x Now, we’ll use algebra to solve for x. Example 2 – Find the value of x. Solution – sin 24° = 12 / x x (sin 24°) = 12 x = 12 / sin 24° x = 29.5

You Try 1 2

Solving Multi-step Trig Problems When two right triangles are combined and we must find the length of a side of one of the triangles, we have to use multiple steps of trig. We always start with triangle that does not contain x to find the relevant information to help us find x. Since both triangles share a side “y”, we need to find the length of that side. Then we'll use that side to help us find the value of x. Example 1 – Find the value of x.

Solving Multi-step Trig Problems (Cont’d) tan 29° = 35 / y y = 35 / tan 29° y = 63.1 tan 67° = 63.1 / x x = 63.1 / tan 67° x = 26.8 y

Solving Multi-step Trig Problems (Cont’d) Example 2 – Find the value of x. Solution – cos 44° = y / 40 y = 40 (cos 44°) y = 28.8 sin 70° = 28.8 / x x = 28.8 / sin 70° x = 30.6 y

You Try Find the value of x.

Using Trig to Find the Area of Triangles Remember that the area of a triangle is ½ x base x height. So we have to find the length of the whole base of the triangle and the height. We’ll let “y” be the height of triangle and “z” be the length of part of the base. “x” is the length of the other part of the base. Next we’ll use trig to solve for x, y, and z. Finally, we’ll plug the values for x, y, and z into the area of a triangle formula. Example 1 - Find the area of the triangle.

Using Trig to Find the Area of Triangles (Cont’d) y z Solution – sin 65° = y / 32 y = 32(sin 65°) y = 29.0 cos 65° = z / 32 z = 32(cos 65°) z = 13.5 tan 35° = x / 29 x = 29(tan 35°) x = 20.3 A = ½ (base)(height) A = ½ ( )(29.0) A = square units

You Try Find the area of the triangle.