Trigonometry in the real world Trigonometry is used by architects, engineers, astronomers, crime scene investigators, flight engineers and many others.

Trigonometry in high school In trigonometry we learn about the sine function, tangent function, and cosine function. These trig functions are abbreviated as sin, cos, and tan. We can use these to calculate sides and angles in right angled triangles. Later, students will be applying this to a variety of situations as well as learning the exact values of sin, cos, and tan for certain angles. Students learn about trigonometric ratios: the law of sines, law of cosines, a new formula for the area of a triangle and applying trigonometric theorems to 3D shapes. Trigonometry for more senior high school students will introduce the reciprocal trig functions, cotangent, secant and cosecant, but you don’t have to worry about these right now! How to answer trigonometry questions The way to answer trigonometry questions depends on whether it is a right angled triangle or not. How to answer trigonometry questions: right angled triangles If your trigonometry question involves a right angled triangle, you can apply the following relationships, ie SOH, CAH, TOA sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent The acronym SOH CAH TOA is used so that you can remember which ratio to use. To answer the trigonometry question: Establish that it is a right angled triangle. Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle). Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle). trigonometry questions step 2 3. Use the following triangles to help us decide which calculation to do: trigonometry questions step 3 3. Use the following triangles to help us decide which calculation to do: trigonometry questions step 3 How to answer trigonometry questions: non-right triangles If the triangle is not a right angled triangle then we need to use the sine rule or the cosine rule. There is also a formula we can use for the area of a triangle, which does not require us to know the base and height of the triangle. Sine rule: a sin ⁡ ( A ) = b sin ⁡ ( B ) = c sin ⁡ ( C ) sin(A) a ​ = sin(B) b ​ = sin(C) c ​ Cosine rule: a 2 = b 2 + c 2 − 2 b c cos ⁡ ( A ) a 2 =b 2 +c 2 −2bccos(A) Area of a triangle: Area = 1 2 a b sin ⁡ ( C ) = 2 1 ​ absin(C) To answer the trigonometry question: Establish that it is not a right angled triangle. Label the sides of the triangle using lowercase a, b, c. Label the angles of the triangle using upper case A, B and C. Opposite sides and angles should use the same letter, for example, angle A is opposite to side a. trigonometry questions step 4 Trigonometry questions In high school geometry, trigonometry questions focus on the understanding of sin, cos, and tan (SOHCAHTOA) to calculate missing sides and angles in right triangles. Trigonometry questions: missing side 1. A zip wire runs between two posts, 25 m 25m apart. The zip wire is at an angle of 10 ∘ 10 ∘ to the horizontal. Calculate the length of the zip wire. trigonometry questions ks3 question 1 25.4 m 25.4m 144.0 m 144.0m 141.8 m 141.8m 24.6 m 24.6m 2. A surveyor wants to know the height of a s2. A surveyor wants to know the height of a skyscraper. He places his inclinometer on a tripod 1 m 1m from the ground. At a distance of 50 m 50m from the skyscraper, he records an angle of elevation of 82 ∘ 82 ∘ . What is the height of the skyscraper? Give your answer to one decimal place. trigonometry questions ks3 question 2 355.8 m 355.8m 7.0 m 7.0m 356.8 m 356.8m 49.5 m 49.5m 3. Triangle ABC is isosceles. Calculate the height of triangle ABC. trigonometry questions ks3 question 3 6 c m 6cm 17.4 c m 17.4cm 34.9 c m 34.9cm 2.1 c m 2.1cm Trigonometry questions: missing angles 4. A builder is constructing a roof. The wood he is using for the sloped section of the roof is 4 m 4m long and the peak of the roof needs to be 2 m 2m high. What angle should the piece of wood make with the base of the roof? trigonometry questions ks3 question 4 26.6 ∘ 26.6 ∘ 60 ∘ 60 ∘ 0.008 ∘ 0.008 ∘ 30 ∘ 30 ∘5. A ladder is leaning against a wall. The ladder is 1.8 m 1.8m long and the bottom of the ladder is 0.5 m 0.5m from the base of the wall. To be considered safe, a ladder must form an angle of between 70 ∘ 70 ∘ and 80 ∘ 80 ∘ with the floor. Is this ladder safe? trigonometry questions ks3 question 5 Yes No Not enough informatio 6. A helicopter flies 40 k m 40km east followed by 105 k m 105km south. On what bearing must the helicopter fly to return home directly? trigonometry questions ks3 question 6 201 ∘ 201 ∘ 159 ∘ 159 ∘ 339 ∘ 339 In geometry, trigonometry questions ask students to solve a variety of problems including multi-step problems and real-life problems. We also need to be familiar with the exact values of the trigonometric functions at certain angles. We look at applying trigonometry to 3D problems as well as using the sine rule, cosine rule, and area of a triangle. Trigonometry questions: SOHCAHTOA 7. Calculate the size of angle ABC. Give your answer to 3 significant figures. trigonometry questions ks4 question 7 24.6 ∘ 24.6 ∘ 38.4 ∘ 38.4 ∘ 18.8 ∘ 18.8 ∘ 21.8 ∘ 21.8 ∘ 8. Kevin’s garden is in the shape of an isosceles trapezoid (the sloping sides are equal in length). Kevin wants to buy enough grass seed for his garden. Each box of grass seed covers 15 m 2 15m 2 . How many boxes of grass seed will Kevin need to buy? trigonometry questions ks4 question 8 6 6 4 4 5 5 10 10 Trigonometry questions: 3D trigonometry 11. Work out angle a, between the line AG and the plane ADHE. trigonometry questions ks4 question 11 14.3 ∘ 14.3 ∘ 15.6 ∘ 15.6 ∘ 15.9 ∘ 15.9 ∘ 90 ∘ 90 ∘ ∘ How to answer trigonometry questions The way to answer trigonometry questions depends on whether it is a right angled triangle or not. How to answer trigonometry questions: right angled triangles If your trigonometry question involves a right angled triangle, you can apply the following relationships, ie SOH, CAH, TOA sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent The acronym SOH CAH TOA is used so that you can remember which ratio to use. To answer the trigonometry question: Establish that it is a right angled triangle. Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle).