Write an expression for \cos \theta using the cosine rule for the following triangle:
Consider the triangle given below:
Consider the triangle given below:
Given \triangle ABC consists of angles A, B and C which appear opposite sides a, b and c respectively:
Consider the following triangle:
Find an expression for a^{2} by using Pythagoras' theorem in \triangle BCD.
Find an expression for h^{2} by using Pythagoras' theorem in \triangle ACD.
Find an expression for x in terms of \cos A.
Substitute your expressions for h^{2} and x into your expression for a^{2} to prove the cosine rule.
To use the cosine rule to find the length ofAC, which angle would need to be given?
Find the length of the missing side in each of the following triangles using the cosine rule. Round your answers to two decimal places.
In \triangle ABC, \cos C = \dfrac{8}{9}:
Find the exact length of side AB in centimetres.
In \triangle QUV, q = 5, u = 6 and \cos V = \dfrac{3}{5}. Find the value of v.
In \triangle QUV, v = 6, q = 10 and u = 12. Find the value of \cos U.
For each of the following triangles, find the value of the pronumeral in degrees. Round your answers to two decimal places.
For each of the following triangles, find \theta to the nearest degree:
Find the value of \theta in the following triangle. Round your answer to the nearest hundredth of a degree.
Find the value of B in the following triangle. Round your answer to two decimal places.
A teacher is writing exam questions for her maths class. She draws a triangle, labels the vertices A, B and C and labels the opposite sides a = 5, b = 8 and c = 15 respectively.
She wants to ask students to find the size of \angle A. Explain why there is an error with her question.
A triangle has sides of length 13 \text{ cm}, 15 \text{ cm} and 5 \text{ cm}. Find the value of x, the largest angle in the triangle to the nearest degree.
The sides of a triangle are in the ratio 4:5:8. Find \theta, the smallest angle in the triangle to the nearest degree.
Eileen is going for a bike ride. She travels 17\text{ km} path from point A to point B. She then turns onto a path at an angle of 115 \degree from her first path and rides for 22\text{ km} until she reaches point C.
Find the value of x, the distance Eileen is from her starting position, correct to two decimal places.
While playing school football, Luke kicks the ball towards the 2\text{ m} wide goal when he is 2.9\text{ m} from one post and 3.3\text{ m} from the other post.
Find the size of the angle, x, in which Luke can score a goal, correct to the nearest degree.
A pendulum of length 76\text{ cm} swings a horizontal distance of 44\text{ cm}.
Find the angle x of the pendulum's movement. Round your answer to the nearest degree.
Dave leaves town along a road on a bearing of 169 \degree and travels 26 \text{ km}. Maria leaves the same town on another road with a bearing of 289 \degree and travels 9 \text{ km}.
Find the direct distance between Dave and Maria. Round your answer to the nearest kilometre.
A pendulum of length 82 \text{ cm} swings a horizontal distance of 31 \text{ cm}.
Find the angle x of the pendulum's movement. Round your answer to the nearest degree.
Lucy is a videographer and has recently bought a drone to capture aerial footage. She wants to test how fast the drone can fly and so flies the drone in a straight line directly above her. According to the readings on the remote control, the drone starts 2.8\text{ m} away from Lucy and after 1.3\text{ s}, it is 4.6\text{ m} away. The angle difference between where the drone started and finished is 118 \degree.
Find the distance covered by the drone, correct to two decimal places.
Find the average speed of the drone in metres per second, correct to two decimal places.
At 4 o'clock, the minute hand points to 12 and the hour hand points to 4. On Michael's clock, the minute hand is 87\text{ mm} and the hour hand is 41\text{ mm} long.
Calculate the value of \theta.
Hence, find the distance, x between the ends of each hand to the nearest millimetre.
The ball is kicked out of bounds in a soccer match and Pascal's team gets to throw the ball back onto the field. His teammate, Eraviste, throws the ball so that Pascal can receive the ball while running up the field.
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When the ball is thrown, Pascal is 12.6\text{ m} away from Eraviste and the angle between him, Eraviste and the path of the ball is 56 \degree. The ball will hit the ground 13.1\text{ m} away from Eraviste.
Find the distance, y, that Pascal needs to cover to reach the ball when it hits the field, correct to one decimal place.
If it takes 1.6\text{ s} for the ball to hit the field after it is thrown, how fast will Pascal have to run to receive the ball in time? Write your answer in metres per second, correct to one decimal place.
While packing away her family’s belongings to move house, Peter drops an expensive photo frame. When she examines the frame, she notices that there are several cracks running in straight lines across the glass.
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She makes the following diagram so that she can find out from a friend how much it will cost to fix:
Find the value of x, correct to one decimal place.
Find the value of k, correct to one decimal place.
If the cost to fix the cracks is \$0.85 for each centimetre of damage, how much will Peter need to pay to get the frame completely fixed?
Edward uses a regular pentagon with sides of 8\text{ cm} each to draw a five pointed star by drawing straight lines between each of the pentagon’s vertices. He then erases the unnecessary lines.
Find the angle \theta between two of the star's points.
Hence, use the cosine rule to find the length of one of the star’s edges, x, correct to two decimal places.
Three fishing spots are shown in the following diagram:
Find the size of the angle \theta to the nearest degree.
Find the other two angles in the triangle formed at Sharpton and Wavemeet.
Hence, find the compass bearing from Troutberk to each of the other fishing spots.
The annual camp for senior students at Harbrook High School is a long hike through a national park. On the last day of the hike, students need to travel from their campsite to the nearby train station.
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There is a path going directly east that they could use to get to the train, but they instead decide to visit a couple of landmarks on their way and, after reading some nearby signs, their teacher draws the following map of the route they will need to take to see each landmark:
Find the true bearing from the campsite (C) to the waterfall (W), correct to the nearest degree.
Find the true bearing from the waterfall to the rest stop, correct to the nearest degree.
Find the direct distance, d, between the rest stop (R) and the train station (T), correct to the nearest metre.
If the students hiked directly east to get to the train station without seeing any landmarks, how far would they have traveled?
If the students walk at an average speed of 0.8\text{ m/s}, find how much longer it would take to visit all the landmarks on the way to the station, compared to only walking directly to the train station from the campsite. Round your answer to the nearest minute.