Right Angled Triangle and Trigonometry
Right Triangle Trigonometry
Trigonometry is a branch of mathematics involving the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Also the ability to use and manipulate trigonometric functions is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Right-angled Triangles In a right-angled triangle the three sides are given special names. The side opposite the right angle is called the hypotenuse (h) – this is always the longest side of the triangle. The other two sides are named in relation to another known angle (or an unknown angle under consideration).
If this angle is known or under consideration
h
θ
this side is called the opposite side because it is opposite the angle
This side is called the adjacent side because it is adjacent to or near the angle Trigonometric Ratios In a right-angled triangle the following ratios are defined sin θ = opposite side length o = hypotenuse length h cosineθ = adjacent side length a = hypotenuse length h
tangentθ =
opposite side length o = adjacent side length a
where θ is the angle as shown
These ratios are abbreviated to sinθ, cosθ, and tanθ respectively. A useful memory aid is Soh Cah Toa pronounced ‘so-car-tow-a’
Page 1 of 5
Unknown sides and angles in right angled triangles can be found using these ratios. Examples Find the value of the indicated unknown (side length or angle) in each of the following diagrams. (1) Method 1. Determine which ratio to use. 2. Write the relevant equation. 3. Substitute values from given information. 4. Solve the equation for the unknown.
b
27o
42
In this problem we have an angle, the opposite side and the adjacent side. The ratio that relates these two sides is the tangent ratio. tan θ = opposite side adjacent side
Substitute in the equation: (opposite side = b, adjacent side = 42, and θ = 27o)
b 42 b = 42 × tan 27° b = 21.4 tan 27° =
Trigonometry is a branch of mathematics involving the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Also the ability to use and manipulate trigonometric functions is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Right-angled Triangles In a right-angled triangle the three sides are given special names. The side opposite the right angle is called the hypotenuse (h) – this is always the longest side of the triangle. The other two sides are named in relation to another known angle (or an unknown angle under consideration).
If this angle is known or under consideration
h
θ
this side is called the opposite side because it is opposite the angle
This side is called the adjacent side because it is adjacent to or near the angle Trigonometric Ratios In a right-angled triangle the following ratios are defined sin θ = opposite side length o = hypotenuse length h cosineθ = adjacent side length a = hypotenuse length h
tangentθ =
opposite side length o = adjacent side length a
where θ is the angle as shown
These ratios are abbreviated to sinθ, cosθ, and tanθ respectively. A useful memory aid is Soh Cah Toa pronounced ‘so-car-tow-a’
Page 1 of 5
Unknown sides and angles in right angled triangles can be found using these ratios. Examples Find the value of the indicated unknown (side length or angle) in each of the following diagrams. (1) Method 1. Determine which ratio to use. 2. Write the relevant equation. 3. Substitute values from given information. 4. Solve the equation for the unknown.
b
27o
42
In this problem we have an angle, the opposite side and the adjacent side. The ratio that relates these two sides is the tangent ratio. tan θ = opposite side adjacent side
Substitute in the equation: (opposite side = b, adjacent side = 42, and θ = 27o)
b 42 b = 42 × tan 27° b = 21.4 tan 27° =