Trigonometry and Triangles review

Two watchtowers are located 20 miles apart. The watchman at tower A sees a tornado at 45 degrees. The watchman at tower B sees the same tornado at 55 degrees. Find the distance from tower A to the tornado.

Draw a picture with the points A and B 20 miles apart. Make the angle at A 45 degrees and the angle at B 55 degrees. The angle at the tornado can be found because the angles of a triangle add to 180. The third angle is 180 - (45 + 55) = 80 degrees. Set up the Law of Sines by the side over the sine of the angle opposite. Therefore, 20 is over the sine of 80 and the unknown side (A to tornado) is over the sine of 55. Solve for the unknown side. x = 20sin55/sin80 = 16.64 miles.

What formula is used to solve "Is it a triangle?"

a + b > c yes a + b ≤ c no

Find the missing sides opposite the 45 and 90 degree angles when the side opposite a 45 degree angle is 5.

b = 5 (angles are equal therefore sides are equal) c = 5 * √2 = 5√2

Find the missing sides opposite the 60 and 90 degree angles when the side opposite the 30 degree angle is 5.

c = 5 * 2 = 10 b = 5 * √3 = 5√3

Is it a triangle? sides are 25, 12, 13

no 12 + 13 = 25, 25 = 25

Is it a triangle? sides are 3, 7, 5

yes 3 + 5 = 8, 8 > 7

What kind of triangle is it? sides are 12, 16, 20

12 * 12 = 144, 16 * 16 = 256, 20 * 20 = 400 144 + 256 = 400 400 = 400 equal so right triangle

What kind of triangle is it? sides are 9, 15, 20

9 * 9 = 81, 15 * 15 = 225, 20 * 20 = 400 81 + 225 = 306 306 < 400 less than so obtuse triangle

For a right triangle, what is the process for finding a side given one side and an angle?

Determine the opposite, adjacent and hypotenuse of the triangle based on the angle given. Determine which trig function relates the side given and the unknown side. Set up the ratio for the trig function using the angle and side. Solve for the missing side.

For a right triangle, what is the process for finding an unknown angle?

Determine the opposite, adjacent and hypotenuse of the triangle based on the angle wanted. Determine which trig function relates the two sides. Set up the ratio for the trig function using the two sides. Solve for the missing angle using the inverse of that function. (second button then trig function) Be sure to label it degrees.

Determine the distance across the river given the following information. The first spot is marked directly across from a rock on the beach. The second spot is 5 feet down the river. The angle from the second spot to the rock is 35 degrees.

Draw a picture. Directly across means a 90 angle at the first spot. The 35 degree angle is given for the second spot. The distance between the first and second spots is 5 feet. The 5 feet is adjacent to the 35 degree angle. The unknown side (the distance across the river) is opposite the 35 degree angle. Opposite and adjacent make the ratio for tangent. Therefore, tan 35 = x divided by 5. x is on top so multiply and x = 5 tan 35. x = 3.5 ft

A television is measured by its diagonal length. What size of television has a height of 21.1 inches and a width of 37.5 inches?

Given two legs. Find the hypotenuse by squaring 21.1 and 37.5 and then adding those two values. 445.21 + 1406.25 = 1851.46. Take the square root of that amount to find the diagonal length. √1851.46 = 43.03 inches. The size of the television 43 inches.

What is the Pythagorean Theorem?

It can only be used with RIGHT triangles

To find an unknown angle given 3 sides and no angles, what equation is used?

Law of Cosines Label the sides so that the side opposite the unknown angle is a. Solve the equation using a as the side opposite the unknown angle and b and c as the other two sides. Solve for the missing angle using the inverse cosine function. (second button then cos) Be sure to label it degrees.

To find a missing side given 2 sides and 1 angle, what equation is used?

Law of Cosines, the missing side must be opposite a known angle.

To find a missing side given a side and 2 angles, what equation is used?

Law of Sines

To find an unknown angle given 2 sides and 1 angle, what equation is used?

Law of Sines Set up the first ratio as the sin of the known angle over the side opposite that angle. Set up the second ratio as the sine of the unknown angle over the side opposite that angle. Get the sine of the unknown angle on one side of the equation by multiplying by the side. Then solve for the missing angle using the inverse sine function. (second button then sin) Be sure to label it degrees.

Given two sides of a right triangle, what equation is used to find the third side?

Pythagorean theorem is the simplest. Label the triangle a, b, and c, where c is the hypotenuse and a and b are legs. Put values into the equation. Add or subtract to find the missing side. Take the square root. Law of cosines can be used if the missing side is the hypotenuse or another angle is known. Law of sines can be used if two angles are known.

Triangle Inequality theorem

The largest side is opposite the largest angle. The smallest side is opposite the smallest angle.