Trigonometric ratios: find an angle measure

Learning Objectives Identify the appropriate inverse trigonometric ratio (sin⁻¹, cos⁻¹, tan⁻¹) based on the given sides of a right-angled triangle. Use a scientific calculator to find the measure of an angle given its sine, cosine, or tangent ratio. Set up a trigonometric equation to represent the relationship between known side lengths and an unknown angle in a right-angled triangle. Solve for an unknown angle in a right-angled triangle using inverse trigonometric functions. Apply the concept of inverse trigonometric ratios to solve real-world problems involving angles of elevation and depression. Round angle measures to a specified degree of accuracy, such as the nearest degree or tenth of a degree. Ever wondered how a skateboard ramp designer calculates the perfect angle...

TermDefinitionExample Inverse Trigonometric FunctionsFunctions that 'undo' the standard trigonometric functions. They take a ratio of side lengths as input and give an angle measure as output.If we know sin(30°) = 0.5, the inverse sine function tells us that sin⁻¹(0.5) = 30°. Inverse Sine (sin⁻¹ or arcsin)The inverse function of sine. It is used to find the measure of an angle in a right-angled triangle when the lengths of the opposite side and the hypotenuse are known.If the opposite side is 5 and the hypotenuse is 10, the angle θ = sin⁻¹(5/10) = 30°. Inverse Cosine (cos⁻¹ or arccos)The inverse function of cosine. It is used to find the measure of an angle in a right-angled triangle when the lengths of the adjacent side and the hypotenuse are known.If the adjacent side is 6 and...

Inverse Sine Formula If \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}, then \theta = \sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right) Use this when you know the lengths of the side opposite the angle you are looking for and the hypotenuse. Inverse Cosine Formula If \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}, then \theta = \cos^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right) Use this when you know the lengths of the side adjacent to the angle you are looking for and the hypotenuse. Inverse Tangent Formula If \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}, then \theta = \tan^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right) Use this when you know the lengths of the side opposite and the side adjacent to...