The basics of trigonometry, including the definitions of functions and basic identities, begin to be studied in geometry classes starting in the 8th grade and continue into the 9th grade. This branch of mathematics lies at the intersection of algebra and geometry and is one of the most complex topics in the school curriculum. The greatest challenge is memorizing many trigonometric formulas. Mathema has prepared an article with basic trigonometric formulas, trigonometric functions of angles, trigonometric identities, and other useful materials.
What You Need to Know to Understand Trigonometric Formulas
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Mathematicians use trigonometric formulas to calculate angles.
- Sine — In a right triangle, the sine of an acute angle is defined as the ratio of the opposite leg to the hypotenuse.
- Cosine — In a right triangle, the cosine of an acute angle is defined as the ratio of the adjacent leg to the hypotenuse.
- Tangent — This is the ratio of the length of the opposite leg to the length of the adjacent leg.
- Cotangent — This is the ratio of the length of the adjacent leg to the length of the opposite leg.
Basic Trigonometric Formulas:
These formulas are essential for solving trigonometric problems in the school curriculum. Here, the most common formulas are collected, along with a table of trigonometric functions for certain angles.
Relationships Between Trigonometric Functions of the Same Argument
\[\sin^2a+\cos^2a\;=\;1\] \[tg\;a=\frac{\sin\;a}{\cos\;a}\] \[tg\;a\;\cdot ctg\;a\;=\;1\] \[1+tg^2a=\frac1{\cos^{2\;}a}\] \[1+ctg^2a=\frac1{\sin^{2\;}a}\]Addition Formulas
\[\sin(a\pm b)=\sin a\cdot\cos b\pm\cos a\cdot\sin b\] \[\cos(a\pm b)=\mathrm{cosa}\cdot\cos b\pm\sin a\cdot\mathrm{sinb}\] \[tg(a\pm b)=\frac{tg\;a+tg\;b}{1\pm tg\;a\cdot tg\;b},\;a\neq\frac\pi2+\pi n,\;b\neq\frac\pi2+\pi n,\;a\pm b\neq\frac\pi2+\pi n,\;n\in\mathbb{Z}\] \[ctg\;(a\;\pm b)\;=\;\frac{ctg\;a\cdot ctg\;b\pm1}{ctg\;a\pm ctg\;b},a\neq\pi n,\;b\neq\pi n,\;a\pm b\neq\pi n,\;n\in\mathbb{Z}\]Double-Angle Formulas
\[\sin2a=2\sin a\cdot\cos a\]
\[\cos2a=\cos^2a-\sin^2a\]
\[tg\;2a\;=\;\frac{2tg\;a}{1-\;tg^2a}\]
\[ctg\;2a\;=\;\frac{ctg^2\;a-1}{2tg\;a}\]
Triple-Angle Formulas
\[\sin3a=3\sin a-4\sin^3a\] \[\cos3a=4\cos^3a-3\cos a\] \[tg3a=\frac{3tg\;a-tg^3a}{1-3tg^2a}\] \[ctg3a=\frac{3ctg\;a-ctg^3a}{1-3ctg^2a}\]Power Reduction Formulas
\[\sin^2a=\frac{1-\cos2a}2\]
\[\cos^2a=\frac{1+\cos2a}2\]
Formulas Converting Products of Trigonometric Functions into Sums
\[\sin a\cdot\sin b=\frac12(\cos(a-b)-(\cos(a+b))\]
\[\cos a\cdot\cos b=\frac12(\cos(a-b)+(\cos(a+b))\]
\[\sin a\cdot\cos b=\frac12(\sin(a-b)+(\sin(a+b))\]
Half-Angle Formulas
\[sina2=±1-cosa2\] \[\cos\frac a2=\pm\sqrt{\frac{1+\cos a}2}\] \[tg\frac a2=\pm\sqrt{\frac{1-\cos a}{1+\cos\;a}}=\frac{\sin a}{1+\cos a},\;a\neq\pi+2\pi n,\;n\in\mathbb{Z}\] \[tg\frac a2=\frac{1-\cos a}{\sin a},\;a\neq\pi n,\;n\in\mathbb{Z}\] \[ctg\frac a2=\pm\sqrt{\frac{1+\cos a}{1-\cos a}}=\;\frac{\sin a}{1-\cos a},\;a\neq2\pi n,\;n\in\mathbb{Z}\] \[ctg\frac a2=\frac{1+\cos a}{\sin a}\;,\;a\neq\pi n,\;n\in\mathbb{Z}\] \[\sin^2\frac a2=\frac{1-\cos\alpha}2\] \[\cos^2\frac\alpha2=\frac{1+\cos\alpha}2\] \[tg^2\frac\alpha2=\frac{1-\cos\alpha}{1+\cos\alpha}\] \[сtg^2\frac\alpha2=\frac{1+\cos\alpha}{1-\;\cos\alpha}\]Trigonometric Table for Specific Angles
This table includes the values of sines, cosines, tangents, and cotangents for certain angles commonly encountered in school trigonometry. It is beneficial to keep this table accessible or to memorize its values. Memorization is more feasible than it might seem, as some values repeat. For example, the tangent of 30° is equal to the cotangent of 60°, and conversely, the tangent of 60° is equal to the cotangent of 30°.
| t | 0 | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| sin t | 0 | 12 | 2‾√2 | 3‾√2 | 1 | 0 | -1 | 0 |
| cos t | 1 | 3‾√2 | 2‾√2 | 12 | 0 | -1 | 0 | 1 |
| tg t | 0 | 3‾√3 | 1 | 3‾√ | – | 0 | – | 0 |
| ctg t | – | 3‾√ | 1 | 3‾√3 | 0 | – | 0 | – |
What Does Trigonometry Explore?
Trigonometry is a branch of mathematics focused on the relationships between the angles and sides of triangles. The word “trigonometry” derives from the Greek words “tría” (three) and “gonía” (angle). Trigonometric functions help us understand how changing the angles of a triangle affects its sides. This enables us to tackle various challenges, such as measuring distances, building heights, and angles on a map, among others.
For example, with a right triangle where one angle and the length of one side are known, trigonometry can be used to determine the length of another side.
Trigonometry is widely applicable in various scientific fields such as engineering, astronomy, physics, computer graphics, and medicine. Learning trigonometry enhances logical thinking and mathematical skills, enabling you to solve diverse problems related to the angles and sides of triangles.
