Introduction to Trigonometry

The One Thing

Trigonometry relates the sides and acute angles of a right-angled triangle through six ratios: sine, cosine, tangent and their reciprocals. Standard-angle values and trigonometric identities then provide methods for calculating unknown quantities and simplifying or verifying expressions.

Definitions and Results

• Right-angled triangle: A triangle containing one angle equal to $90^{\circ }$.
• Hypotenuse: The side opposite the right angle; it is always the longest side of a right-angled triangle.
• Perpendicular: The side opposite the selected acute angle.
• Base: The side adjacent to the selected acute angle, excluding the hypotenuse.
• Sine ($\sin$): For an acute angle $\theta$,

$$\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}.$$

• Cosine ($\cos$): For an acute angle $\theta$,

$$\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}.$$

• Tangent ($\tan$): For an acute angle $\theta$,

$$\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}.$$

• Cosecant ($\mathrm{cosec}$): The reciprocal of sine:

$$\mathrm{cosec}\theta =\frac{\text{Hypotenuse}}{\text{Perpendicular}}=\frac{1}{\sin \theta }.$$

• Secant ($\sec$): The reciprocal of cosine:

$$\sec \theta =\frac{\text{Hypotenuse}}{\text{Base}}=\frac{1}{\cos \theta }.$$

• Cotangent ($\cot$): The reciprocal of tangent:

$$\cot \theta =\frac{\text{Base}}{\text{Perpendicular}}=\frac{1}{\tan \theta }.$$

• Trigonometric identity: An equality involving trigonometric ratios that is true for every angle for which both sides are defined. Denominators must not be zero.
• Complementary angles: Two angles whose sum is $90^{\circ }$. In a right triangle, the ratios of complementary angles are related.
• Quotient relationships:

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }.$$

• Pythagorean identities:

$${\sin }^2\theta +{\cos }^2\theta =1,$$ $$1+{\tan }^2\theta ={\sec }^2\theta ,$$ $$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

• Equivalent Pythagorean forms:

$$1-{\sin }^2\theta ={\cos }^2\theta ,1-{\cos }^2\theta ={\sin }^2\theta ,$$ $${\sec }^2\theta -{\tan }^2\theta =1,{\mathrm{cosec}}^2\theta -{\cot }^2\theta =1.$$

• Complementary-angle relationships:

$$\sin (90^{\circ }-\theta )=\cos \theta ,$$ $$\cos (90^{\circ }-\theta )=\sin \theta ,$$ $$\tan (90^{\circ }-\theta )=\cot \theta ,$$ $$\cot (90^{\circ }-\theta )=\tan \theta ,$$ $$\sec (90^{\circ }-\theta )=\mathrm{cosec}\theta ,$$ $$\mathrm{cosec}(90^{\circ }-\theta )=\sec \theta .$$

• Standard sine values:

$$\sin 0^{\circ }=0,\sin 30^{\circ }=\frac{1}{2},\sin 45^{\circ }=\frac{1}{\sqrt{2}},$$ $$\sin 60^{\circ }=\frac{\sqrt{3}}{2},\sin 90^{\circ }=1.$$

• Standard cosine values:

$$\cos 0^{\circ }=1,\cos 30^{\circ }=\frac{\sqrt{3}}{2},\cos 45^{\circ }=\frac{1}{\sqrt{2}},$$ $$\cos 60^{\circ }=\frac{1}{2},\cos 90^{\circ }=0.$$

• Standard tangent values:

$$\tan 0^{\circ }=0,\tan 30^{\circ }=\frac{1}{\sqrt{3}},\tan 45^{\circ }=1,\tan 60^{\circ }=\sqrt{3}.$$ $\tan 90^{\circ }$ is not defined.

• Standard cosecant values:

$$\mathrm{cosec}{0}^{\circ }\text{ is not defined},\mathrm{cosec}30^{\circ }=2,\mathrm{cosec}45^{\circ }=\sqrt{2},$$ $$\mathrm{cosec}60^{\circ }=\frac{2}{\sqrt{3}},\mathrm{cosec}90^{\circ }=1.$$

• Standard secant values:

$$\sec 0^{\circ }=1,\sec 30^{\circ }=\frac{2}{\sqrt{3}},\sec 45^{\circ }=\sqrt{2},$$ $$\sec 60^{\circ }=2,\sec 90^{\circ }\text{ is not defined}.$$

• Standard cotangent values:

$$\cot 0^{\circ }\text{ is not defined},\cot 30^{\circ }=\sqrt{3},\cot 45^{\circ }=1,$$ $$\cot 60^{\circ }=\frac{1}{\sqrt{3}},\cot 90^{\circ }=0.$$

• Dimensionless ratios: Trigonometric ratios have no units because they are ratios of lengths.
• Dependence on the reference angle: The values of the ratios depend on the selected angle and sides. The names perpendicular and base change when the reference angle changes, but the hypotenuse remains opposite the $90^{\circ }$ angle.

Worked Methods

Selecting and using a trigonometric ratio

• Identify the selected acute angle $\theta$.
• Identify the hypotenuse as the side opposite the $90^{\circ }$ angle.
• Relative to $\theta$, identify the perpendicular as the opposite side and the base as the adjacent side excluding the hypotenuse.
• Choose the ratio according to the known and required sides:

$$\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}},$$ $$\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}},$$ $$\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}.$$

• Use a reciprocal ratio when appropriate:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }.$$

• Solve the resulting equation for the unknown side or angle.

The correct ratio is determined by the known sides and the required side or angle. Trigonometric ratios are dimensionless because they compare lengths.

Using standard-angle values

• Identify whether the angle is $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, or $90^{\circ }$.
• Select the required ratio from the standard-value results.
• Substitute the value into the problem.
• Use a reciprocal relationship where necessary. For example,

$$\mathrm{cosec}30^{\circ }=\frac{1}{\sin 30^{\circ }}=\frac{1}{1/2}=2.$$

• Check whether the ratio is defined. For example, $\tan 90^{\circ }$, $\mathrm{cosec}{0}^{\circ }$, $\sec 90^{\circ }$, and $\cot 0^{\circ }$ are not defined.

These values arise from the geometry of special right triangles and are used to calculate unknown quantities.

Using complementary angles

• Confirm that the two angles have sum $90^{\circ }$.
• Replace the ratio of $90^{\circ }-\theta$ using:

$$\sin (90^{\circ }-\theta )=\cos \theta ,$$ $$\cos (90^{\circ }-\theta )=\sin \theta ,$$ $$\tan (90^{\circ }-\theta )=\cot \theta ,$$ $$\cot (90^{\circ }-\theta )=\tan \theta ,$$ $$\sec (90^{\circ }-\theta )=\mathrm{cosec}\theta ,$$ $$\mathrm{cosec}(90^{\circ }-\theta )=\sec \theta .$$

• Continue the calculation using the resulting ratio.

Proving or simplifying a trigonometric identity

• Begin with one side of the identity, or transform both sides separately.
• Use the definitions of the six ratios, reciprocal relationships, quotient relationships, and Pythagorean identities.
• Simplify until the chosen side becomes the other side or both sides become a common form.
• Ensure that all expressions remain defined; denominators must not be zero.
• Use the Pythagorean identities, which follow from the Pythagorean theorem:

$${\sin }^2\theta +{\cos }^2\theta =1,$$ $$1+{\tan }^2\theta ={\sec }^2\theta ,$$ $$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

For example, the equivalent form $${\sec }^2\theta -{\tan }^2\theta =1$$ is obtained by rearranging $$1+{\tan }^2\theta ={\sec }^2\theta .$$ Similarly, $${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$$ is obtained from $$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

Where It Goes Wrong

• The hypotenuse is incorrectly identified. It must be opposite the $90^{\circ }$ angle and is always the longest side.
• The perpendicular and base are assigned without reference to the selected acute angle. Their names change when the reference angle changes.
• The wrong basic ratio is selected instead of matching the known and required sides to sine, cosine, or tangent.
• Reciprocal ratios are inverted incorrectly: $\mathrm{cosec}\theta =1/\sin \theta$, $\sec \theta =1/\cos \theta$, and $\cot \theta =1/\tan \theta$.
• An identity is verified only for one angle rather than being shown true for every permitted value in its domain.
• Undefined values and zero denominators are overlooked, including $\tan 90^{\circ }$, $\mathrm{cosec}{0}^{\circ }$, $\sec 90^{\circ }$, and $\cot 0^{\circ }$.

What Gets Asked

This material supports questions requiring students to:

• Define the hypotenuse, perpendicular, base, and six trigonometric ratios in a right-angled triangle.
• Select the correct ratio from given sides and calculate an unknown side or angle.
• Use reciprocal and quotient relationships, including

$$\tan \theta =\frac{\sin \theta }{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }.$$

• Recall and apply the standard values for $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, and $90^{\circ }$.
• Determine whether a trigonometric ratio is defined at a given angle.
• Apply complementary-angle relationships.
• Simplify expressions using

$${\sin }^2\theta +{\cos }^2\theta =1,1+{\tan }^2\theta ={\sec }^2\theta ,1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta .$$

• Prove trigonometric identities by reducing one or both sides to a common form.
• Explain why trigonometric ratios are dimensionless and why their values depend on the selected reference angle.