Trigonometry – Right-Angled Triangles - Finding an angle

Trigonometry – Right-Angled Triangles - Finding an angle

Understanding Right-Angled Triangle Trigonometry

  • Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

  • In a right-angled triangle, one angle is exactly 90 degrees.

  • The two sides forming the right angle are the adjacent (next to the angle) and the opposite (across from the angle). The longest side, across from the right angle, is the hypotenuse.

  • There are three basic trigonometric ratios used to find the angle of a right-angled triangle: sine (sin), cosine (cos), and tangent (tan).

Trigonometric Ratios

  • The sine ratio (sin) of an angle in a right-angled triangle is defined as the length of the opposite side divided by the length of the hypotenuse.

  • The cosine ratio (cos) of an angle is the length of the adjacent side divided by the length of the hypotenuse.

  • The tangent ratio (tan) of an angle is the length of the opposite side divided by the length of the adjacent side.

Finding an Angle Using Trigonometric Ratios

  • You can find an angle in a right-angled triangle if you know the lengths of two of its sides.

  • Use the correct trigonometric ratio based on which two sides you know.

  • If you know the lengths of the opposite side and the hypotenuse, use the sine ratio. Knowing the lengths of the adjacent side and the hypotenuse calls for the cosine ratio. If you know the lengths of the opposite and adjacent sides, use the tangent ratio.

  • To find the angle, use the inverse trigonometric functions (sin^-1, cos^-1, tan^-1) on a calculator.

  • The inverse trigonometric function will give the size of the angle that has the given sine, cosine or tangent ratio.

Practical Tips

  • It’s important to keep the sides straight - know which is the hypotenuse, adjacent, and opposite relative to your target angle!

  • Always read questions carefully and draw a labelled diagram to visualise the triangle.

  • Remember to check your calculator is set to degrees (DEG), not radians (RAD), before calculating.

  • Practicing with different examples is key to mastering finding angles in right-angled triangles.