Unit 6 Review PDF - Trigonometry Problems

# Unit 6 Review ## Similar Triangles 1. Determine all missing lengths in the two triangles. - Triangle 1: - **M** - **Q** - **N** - **Y** - **MQ** = 9 cm - **QN** = 5 cm - **NY** = 18 cm - Triangle 2: - **M** - **O** -...

# Unit 6 Review ## Similar Triangles 1. Determine all missing lengths in the two triangles. - Triangle 1: - **M** - **Q** - **N** - **Y** - **MQ** = 9 cm - **QN** = 5 cm - **NY** = 18 cm - Triangle 2: - **M** - **O** - **X** - **Y** - **MO** = 9 cm - **OX** = 12 cm - **XY** = 18 cm ## Primary Trigonometric Ratios (sin, cos, tan) 2. Determine the measure of angle θ, to the nearest degree. **(a)** - A triangle with vertices A, B and C. - **AB** = 10 cm - **BC** = 8 cm - **AC** is the hypotenuse. - Angle θ is at vertex C. **(b)** - A triangle with vertices D, E and F. - **DF** = 21 cm - **DE** = 20 cm - **EF** is the hypotenuse. - Angle θ is at vertex D. **(c)** - A triangle with vertices P, Q and R. - **PQ** = 24 cm - **PR** = 25 cm - **QR** is the hypotenuse. - Angle θ is at vertex Q. ## Other Problems involving Right Triangles 3. Determine the value of x, to the nearest tenth. **(a)** - A triangle with vertices A, B and C. - **AB** = 36 cm - **BC** = x - **AC** is the hypotenuse. - Angle at vertex B = 38° **(b)** - A triangle with vertices D, E and F. - **DE** = 41 cm - **DF** = x - **EF** is the hypotenuse. - Angle at vertex D = 83° 4. Solve the following triangles, to the nearest tenth for lengths and the nearest degree for angles. **(a)** - A triangle with vertices K, L and M. - **KL** = 24 mm - **KM** = 46 mm - **LM** is the hypotenuse. **(b)** - A triangle with vertices R, S and T. - **RS** = 17 km - **RT** is the hypotenuse. - Angle at vertex S = 55° 5. Find x to the nearest tenth, then find y to the nearest degree. - A triangle with vertices: - **X** - **Y** - **Z** - Angle at Z = 48° - **XZ** = 8 cm - **YZ** = 9 cm - **XY** = x - Angle at Y = y 6. Find the length of CD, to the nearest tenth. - A triangle with vertices: - **A** - **B** - **C** - **D** - Angle at A = 90° - Angle at B = 43.4° - Angle at C = 34° - **AB** = 100 cm - **CD** is one of the legs. 7. Find the length of OP, to the nearest tenth. - A rectangle with vertices: - **O** - **P** - **Q** - **R** - **OQ** = 50 m - **QR** = 40 m - **OP** is the hypotenuse of the right triangle formed by connecting points O, Q and P. - Angle at O = 39.7° - Angle at Q = 50.3° ## Problem Solving using Right Triangles 8. A radio transmitter is to be supported with a guy wire as shown. The wire is to form a 65° angle with the ground and reach 30 m up the transmitter. The wire can be ordered in whole-number lengths of meters. How much wire should be ordered to the nearest tenth of a meter? - A right triangle with the following: - **Guy wire** is the hypotenuse. - Angle between guy wire and ground = 65° - **Height of tower** = 30 m - **Length of guy wire** = x 9. A kite string is 35 m long. The angle the string makes with the ground is 50°. To the nearest tenth of a meter, how far from the person holding the string is a person standing directly under the kite? - A right triangle with the following: - **Length of the kite string** = 35 m - Angle between string and ground = 50° - **Distance between people** = x 10. A surveyor is positioned at a traffic intersection, viewing a marker on the other side of the street. The marker is 18 m from the intersection. The surveyor cannot measure the width directly because there is too much traffic. Find the width of James Street, to the nearest tenth of a meter. - A right triangle with the following: - **Distance between the marker and the intersection** = 18 m - **Angle at intersection** = 64° - **Width of James Street** = x ## Other Problems involving Acute Triangles 11. The support for a shelf makes an angle of 48° with the wall. If the shelf is 32 cm wide, what is the length of the support, to the nearest tenth of a centimeter? - A right triangle with the following: - **Length of support** = x - **Angle between support and wall** = 48° - **Width of shelf** = 32 cm 12. The diagram shows the roof of a house. How wide is the house to the nearest tenth of a meter? - An isosceles triangle with the following: - **Rafters** = 3m - **Angle at the top** = 26° - **Width of house** = x 13. Two buildings are separated by a distance of 16 m. From the top of one building, the angle of elevation to the second building is 48°, and the angle of depression to the bottom of the second building is 64°. Calculate the heights of the two buildings, to the nearest tenth of a meter. ## Sine Law and Cosine Law 14. Using sine law, find the length of the indicated side, to the nearest tenth of a unit, or the measure of the indicated angle to the nearest degree. **(a)** - A triangle with vertices A, B, and C. - **AC** = 20 cm - **AB** = 15 cm - Angle at C = 57° - Angle at A = θ - **BC** = a - `c` = `20` - `a` = `15` - `b` = `...` - **A** = 57° - **B** = θ - **C** = ... **(b)** - A triangle with vertices A, B, and C. - **AB** = 19.5 cm - **BC** = a - Angle at A = 68° - Angle at B = 59° - Angle at C = θ 15. Using cosine law, find the length of the indicated side, to the nearest tenth of a unit, or the measure of the indicated angle, to the nearest degree. **(a)** - A triangle with vertices A, B, and C. - **BC** = 7.6 cm - **AC** = 4.3 cm - **AB** = b - Angle at B = 70° - Angle at A = θ **(b)** - A triangle with vertices A, B, and C. - **AB** = 7 cm - **BC** = 11 cm - **AC** = b - Angle at B = θ ## Solving Triangles 16. Solve the following triangles. Round to the nearest tenth for lengths and the nearest degree for angles. **(a)** - A triangle with vertices P, Q, and R. - **PQ** = 11 m - **PR** = 12 m - Angle at Q = 61° **(b)** - A triangle with vertices P, Q, and R. - **PQ** = 16 cm - **PR** = 15 cm - **QR** = 12 cm ## Problem Solving using Acute Triangles 17. A small plane and a jet are 7.5 km from each other. From an observation tower, the aircrafts are separated by an angle of 68°. If the jet is 5.2 km from the tower, how far is the plane from the tower, to the nearest tenth of a km? - A triangle with the following: - **Distance between the plane and the jet** = 7.5 km - **Distance between the jet and the tower** = 5.2 km - **Distance between the plane and the tower** = x - **Angle at the tower** = 68° 18. While standing exactly halfway between two trees, Nikki spots a blue jay at the top of one tree and a cardinal at the top of the other. From her point of view, the angle of elevation to the blue jay is 60° and the angle of elevation to the cardinal is 50°. The trees are 40 m apart. **(a)** If Nikki is 1.5 m tall, find the height of each tree to the nearest tenth of a meter. **(b)** How far apart are the two birds to the nearest tenth of a meter? ## Solutions 1. **q** = 6 cm, **y** = 10 cm 2. **(a)** 37° **(b)** 46° **(c)** 74° 3. **(a)** 45.7 cm **(b)** 40.7 cm 4. **(a)** **k** = 39.2 mm, ∠ **K** = 59°, ∠ **M** = 31° **(b)** ∠ **T** = 35°, **r** = 29.6 km, **s** = 24.3 km 5. **x** = 8.9 cm, **y** = 49° 6. 114.1 cm 7. 90.2 m 8. 33.1 m 9. 22.5 m 10. 8.8 m 11. 43.1 cm 12. 12.3 m 13. **short** = 32.8 m, **tall** = 50.6 m 14. **(a)** 39° **(b)** 55° 15. **(a)** 7.3 cm **(b)** 17.7 cm 16. **(a)** ∠ **Q** = 53°, ∠ **P** = 66°, **p** = 12.5 m **(b)** ∠ **P** = 63°, ∠ **Q** = 72°, ∠ **R** = 45° 17. 7.7 km 18. **(a)** blue jay = 36.1 m, cardinal = 25.3 **(b)** 41.4 m