Geometry

Trigonometric Ratios

Big Picture

When given enough information, trignometric ratios can be used to determine the angle measurements or side lengths of a triangle. Trigonometric functions are used to find the measures of all the sides and angles in a right triangle if given:

  • Two side lengths
  • One side length and the measure of one acute angle

Key Terms

Trigonometry: The study of the relationships between the sides and angles of right triangles.

Trigonometric Ratio: A ratio of the lengths of two sides in a right triangle.

Tangent: For an acute angle θ in a right triangle, tan θ is equal to the ratio of the side opposite to θ over the side adjacent to θ.

Sine: For an acute angle θ in a right triangle, sin  θ is equal to the ratio of the side opposite to θ  over the hypotenuse of the triangle.

Cosine: For an acute angle θ in a right triangle, the cos θ is equal to the ratio of the side adjacent to θ over the side adjacent to θ.

Parts of a Triangle

The hypotenuse is always the side opposite of the right angle. However, adjacent and opposite depend on the angle we are talking about. The adjacent side helps to create the angle, while the opposite side is directly opposite the angle.

Parts of a Triangle

3 Basic Trigonometric Ratios

Trigonometric ratios relate the lengths of two sides in a right triangle. Each ratio is a function of the angle. For a given acute angle θ, the ratios are constant. The three basic ratiosare:

Tangent:  \(\tan \theta = (\frac{\text{opposite}}{\text{adjacent}})\)           Sine: \(\sin \theta = (\frac{\text{opposite}}{\text{hypotenuse}})\)

Cosine:  \(\cos \theta = (\frac{\text{adjacent}}{\text{hypotenuse}})\)

Remember: These relationships only work for right triangles!

A way to remember these ratios is the mnemonic SOH-CAH-TOA:

\(\color{blue}{S}\color{black}{ine = }(\frac{\color{blue}{O}\color{black}{pposite}}{\color{blue}{H}\color{black}{ypotenuse}}) \qquad \color{red}{C}\color{black}{osine = }(\frac{\color{red}{A}\color{black}{djacent}}{\color{red}{H}\color{black}{ypotenuse}}) \qquad \color{green}{T}\color{black}{angent = }(\frac{\color{green}{O}\color{black}{pposite}}{\color{green}{A}\color{black}{djacent}})\)

Another way to remember SOH-CAH-TOA is Some Old Horse, Caught Another Horse, Taking Oats Away.Some other important points:

  • Of the three basic trigonometric functions, only the tangent ratio can be greater than 1.
  • For the other two functions, the length of the hypotenuse is in the denominator. Since the hypotenuse has the longest length, the sine and cosine ratios cannot be greater than 1.
  • If two right triangles are similar, then the sine, cosine, and tangent ratios will be the same.

Geometry

Trigonometric Ratios Cont.

Inverses

The inverses of sine, cosine, and tangent can also be used. Instead of plugging in angles to find side ratios, the inverses are used by plugging in side ratios to find angle measures. Inverses are useful if we know the side lengths of a right triangle and want to find its angle measures.

  • \(sin^{-1} (\frac{opposite}{hypotenuse}) = \theta\)
  • Also called arcsine \(arcsin (\frac{opposite}{hypotenuse}) = \theta\)
  • \(cos^{-1} (\frac{adjacent}{hypotenuse}) = \theta\)
  • Also called arccosine, \(arccos (\frac{adjacent}{hypotenuse}) = \theta\)
  • \(tan^{-1} (\frac{opposite}{adjacent}) = \theta\)
  • Also called arctangent, or \(arctan (\frac{opposite}{adjacent}) = \theta\)

The raised -1 in \(tan^{-1}\), \(sin^{-1}\), and \(cos^{-1}\) is NOT an exponent.

Applications

Angles of Depression and Elevation

A common problem that uses trigonometric functions is determining the angles of depression and elevation.

Angle of depression: The angle measured down from the horizontal line.

Angle of depression

Angle of elevation: The angle measured up from the horizontal line.

Angle of elevation

Area of a Triangle

Now that we know the definitions of sine, cosine, and tangent, we can express the height of a triangle as b sin A:

Area of a Triangle

Since the height of the triangle is b sin A,  the  area  of  the  triangle  is: \(\frac{1}{2}\)(base \(\cdot\) height) = \(\frac{1}{2}cb \sin A = \frac{1}{2}bc sin A\)

  • Since there are three altitudes that can be drawn, the area of the triangle can also be expressed as \(\frac{1}{2}ab \sin C\) or \(\frac{1}{2}ac \sin B\).
  • This formula is useful when two sides and an angle of a triangle are known and we want to find the area. Keep in mind that this formula will work for any triangle, not just for right triangles.

Geometry

Trigonometric Ratios Cont.

Laws of Sines and Cosines

Law of Sines

The law of sines can be used to find missing side lengths or angles in a triangle. The law states that in any triangle, the ratio of the side length to the sine of the opposite angle will be constant. Basically, the ratio is the same for all three angles and their side lengths. The law of sines is shown with the equation:

\((\frac{a}{\sin A})\) = \((\frac{b}{\sin B})\) = \((\frac{c}{\sin C})\)

Use law of sines when given:

  • An angle and its opposite side.
  • Any two angles and one side.
  • Two sides and the non-included angle.
Law of Sines

Law of Cosines

The law of cosines can be used to find a missing side length or a missing angle in a triangle. To use the law of cosines, we must have either the measures of all three sides, or the measure of two sides and the measure of the included angle. The law of cosines is:

\(a^2 = b^2 + c^2 - 2bc(cos A)\)

\(b^2 = a^2 + c^2 - 2ac(cos B)\)

\(c^2 = a^2 + b^2 - 2ab(cos C)\)

Use law of cosines when given:

  • Two sides and the included angle.
  • All three sides.

Notes