Trigonometric Functions, Identities - Exact Values for Special Angles

Trigonometry is the branch of mathematics which studies the relationship between the sides of a right-angled triangle and its angles. It helps us determine angles and length of sides related to the angle. In scientific calculation, trigonometric values are often referred to the table of trigonometric ratios, which are given in rounded decimals, and important for obtaining accurate result. In some calculations, exact values for trigonometric functions of some special angles may be necessary, especially for those that could be represented by real numbers.

Definition of Trigonometric Functions

The definition of the trigonometric functions involves the sides of the right-angled triangle. In a right triangle, the longest side is called hypotenuse, an opposite side is the one "opposite to" a given acute angle, and an adjacent side is the one next to a given acute angle. The trigonometric functions are defined as ratios of these sides in various combinations.

There are six trigonometric functions for an angle, in which sine, cosine, and tangent are the primary ones while cosecant, secant and cotangent are basically reciprocals of relevant primary trigonometric functions. In mathematical calculations, the names of trigonometric functions are the abbreviation forms of their English words, which are usually consistent to trig functions in other system, software, such as Micrsoft Excel formula. Here is the table of trigonometric functions and their definitions.


Trig Name Full Name Definition
sin sine \dfrac{\text{ Opposite}}{\text{Hypotenuse}}
cos cosine \dfrac{\text{Adjacent }}{\text{Hypotenuse}}
tan tangent \dfrac{\text{ Opposite}}{\text{Adjacent}}
csc cosecant \dfrac{\text{ Hypotenuse}}{\text{Opposite}}
sec secant \dfrac{\text{Hypotenuse}}{\text{Adjacent }}
cot cotangent
\dfrac{\text{ Adjacent}}{\text{Opposite}} |

Units of Angles

There are two commonly used units of measurement for angles, degrees and radians. — A degree usually denoted by ° (the degree symbol), is defined as \dfrac{1}{360} of one revolution. In trigonometry, radian is more popularly used. 2π of radians is defined as the ratio of the circumference of the entire circle to radius, i.e., one revolution. So the relationship of degree and radian can be derived as,

1° = \dfrac{π}{180}

Angles could be negative or positive. When an angle is measured counterclockwise, it is positive. Otherwise, when an angle is measured clockwise, it is negative. The sign of a trigonometric function of an angle is related to the size and sign of the angle.

Sine Function

Sin of an angle is defined as the ratio of the opposite side to hypotenuse of a right triangle that contains the angle. In the figure, the value of sin could be expressed as,

\sin \alpha = \dfrac{\text{ Opposite}}{\text{Hypotenuse}} = \dfrac{BC}{AC}

Since the length of opposite side is less than that of hypotenuse, the absolute value of sine function of an angle is always less than or equal to 1. When the angle is equal to 90\degree, which is a degenerate case, \sin90\degree=1.

Cosine Function

Cosine of an angle is the ratio of the adjacent side to the hypotenuse. In the above figure, the cosine of the angle \alpha will be derived as,

\cos \alpha = \dfrac{\text{Adjacent }}{\text{Hypotenuse}} = \dfrac{AB}{AC}

Similar to sine function, the absolute range of a cosine function is also less than or equal to 1.

Tangent Function

The tangent function is the ratio of the opposite side to the adjacent side. It should be noted that the tangent could also be defined in terms of the ratio of sine to cosine. From the figure above, the tangent function of the angle \alpha be expressed as,

\tan \alpha = \dfrac{\sin \alpha }{\cos\alpha } = \dfrac{\text{ Opposite}}{\text{Adjacent}} = \dfrac{BC}{AB}

When AB approaches infinitely small, or zero, \alpha \to 90\degree, \tan\alpha \to \infty. But that will never happen since the denominator can never be zero in fraction. Therefore, 90\degree is not included in the domain of the tangent function.

Cosecant Function

The cosecant function is the ratio of the hypotenuse to the opposite side. From the definition, it could also be represented as the multiplicative inverse of sine function.

\csc \alpha = \dfrac{1}{\sin\alpha } = \dfrac{\text{ Hypotenuse}}{\text{Opposite}} = \dfrac{AC}{BC}

As the opposite of sine function, the absolute value of cosecant function is always larger than or equal to 1. Similar to tangent function, zero degree is not included in the domain of cosecant function.

Secant Function

The secant function is the ratio of the hypotenuse to the adjacent side. It could also be represented as the reciprocal of cosine function.

\sec \alpha = \dfrac{1}{\cos \alpha } =  \dfrac{\text{ Hypotenuse}}{\text{Adjacent}} = \dfrac{AC}{AB}

As the reciprocal of cosine function, the absolute value of secant function is always larger than or equal to 1. Similar to tangent function, 90 degree is not included in the domain of secant function.

Cotangent Function

The cotangent function is the ratio of the adjacent to the adjacent side. It could also be represented as the reciprocal of tangent function.

\cot \alpha = \dfrac{1}{\tan\alpha } = \dfrac{\text{ Adjacent}}{\text{Opposite}} = \dfrac{AB}{BC}

As the reciprocal of tangent function, the absolute value of cotangent function can be infinitely large. But zero degree is not included in the domain of cotangent function.

Trigonometric identities for Special Angles

One of interesting applications using trigonometric functions is to derive the exact values of special angles. Before starting the work, let's have refresh of trigonometric identities we may use to facilitate our process.

Reciprocal Identities

Since sine and cosecant, cosine and secant , tangent and cotangent are pairs of reciprocals, when multiplied together, equal 1, which are known as Reciprocal identities.

\def\arraystretch{2.5} \begin{array}{cc} \sin \alpha = \dfrac{1}{\csc \alpha} &\csc \alpha = \dfrac{1}{\sin \alpha} & \\ \cos \alpha = \dfrac{1}{\sec \alpha} &\sec \alpha = \dfrac{1}{\cos \alpha} & \\ \tan \alpha = \dfrac{1}{\cot \alpha} &\cot \alpha = \dfrac{1}{\tan \alpha} & \\ \end{array}

These identities are useful when calculating values of certain trigonometric functions when their reciprocal ones are known values.

Cofunction Identities

Two trigonometric functions are cofunctions if they are equal on complementary angles. The value of a trigonometric function of an angle equals the value of the cofunction of the complement.There 3 pairs of cofunctions, Sine and cosine, tangent and cotangent, secant and cosecant.

\def\arraystretch{2} \begin{array}{cc} \cos(\dfrac{\pi}{2}-\alpha)=\sin \alpha&\cos(90\degree-\alpha)=\sin \alpha& \\ \sin(\dfrac{\pi}{2}-\alpha)=\cos \alpha&\sin(90\degree-\alpha)=\cos \alpha& \\ \tan(\dfrac{\pi}{2}-\alpha)=\cot \alpha&\tan(90\degree-\alpha)=\cot \alpha& \\ \cot(\dfrac{\pi}{2}-\alpha)=\tan \alpha&\cot(90\degree-\alpha)=\tan \alpha& \\ \sec(\dfrac{\pi}{2}-\alpha)=\csc \alpha&\sec(90\degree-\alpha)=\csc \alpha& \\ \csc(\dfrac{\pi}{2}-\alpha)=\sec \alpha&\csc(90\degree-\alpha)=\sec \alpha& \\ \end{array}

Using cofunction identities reduce half of work load since we only need to consider special angles from 1\degree to 45\degree , or vise verse.

Sum and difference identities

\def\arraystretch{2.2} \begin{array}{ll} \cos(\alpha +\beta ) = \cos \alpha\cos \beta -\sin \alpha \sin \beta & \\ \cos(\alpha -\beta ) = \cos \alpha\cos \beta +\sin \alpha \sin \beta & \\ \sin(\alpha +\beta ) = \sin \alpha\cos \beta +\cos \alpha \sin \beta & \\ \sin(\alpha -\beta ) = \sin \alpha\cos \beta -\cos \alpha \sin \beta & \\ \tan(\alpha+\beta) = \dfrac{\tan \alpha + \tan \beta }{1-\tan \alpha\tan \beta} & \\ \tan(\alpha -\beta) = \dfrac{\tan \alpha - \tan \beta }{1+\tan \alpha\tan \beta} & \\ \cot(\alpha+\beta) = \dfrac{\cos(\alpha +\beta)}{\sin(\alpha +\beta)} = \dfrac{ \cos \alpha\cos \beta -\sin \alpha \sin \beta}{\sin \alpha\cos \beta +\cos \alpha \sin \beta} =\dfrac{\cot \alpha \cot \beta -1 }{\cot \alpha+ \cot \beta } & \\ \cot(\alpha-\beta) =\dfrac{\cot \alpha \cot \beta +1 }{\cot \alpha- \cot \beta } \end{array}

Sum and difference identities could transform a special angle to the form of sum or difference of two angles whose trigonometric values are known values.

Double angle identities

\def\arraystretch{2.5} \begin{array}{ll} \sin2 \alpha = 2\sin \alpha \cos \alpha & \\ \cos2 \alpha =\cos^2\alpha - \sin^2 \alpha & \\ \cos2 \alpha = 2\cos^2\alpha -1 & \\ \cos2 \alpha = 1-2 \sin^2 \alpha & \\ \tan 2 \alpha = \dfrac{2\tan \alpha }{1-\tan^2\alpha } & \\ \cot 2 \alpha = \dfrac{1}{\tan 2\alpha } = \dfrac{ 1-\tan^2\alpha }{ 2\tan \alpha }= \dfrac{ 1-\dfrac{1}{\cot^2\alpha } }{ \dfrac{2}{\cot \alpha } } =\dfrac{\cot^2\alpha -1 }{2\cot \alpha } \\ \sin 2\alpha = 2\sin \alpha \cos \alpha=\dfrac{ 2\sin \alpha \cos \alpha}{\sin^2 \alpha +\cos^2 \alpha } = \dfrac{2\tan \alpha }{1+\tan^2 \alpha } & \\ \cos 2\alpha = \cos^2\alpha - \sin^2 \alpha = \dfrac{\cos^2\alpha - \sin^2 \alpha }{\cos^2 \alpha+\sin^2\alpha } =\dfrac{1-\tan^2 \alpha }{1+\tan^2\alpha } & \\ \end{array}

Double angle identities are useful when the trigonometric functions of its semi angle are known values.

Half angle identities

\def\arraystretch{3} \begin{array}{ll} \sin \dfrac{\alpha }{2}=\pm\sqrt{\dfrac{1-\cos \alpha }{2} } & \\ \cos \dfrac{\alpha }{2}=\pm\sqrt{\dfrac{1+\cos \alpha }{2} } & \\ \tan \dfrac{\alpha }{2}=\pm\sqrt{\dfrac{1-\cos \alpha }{1+\cos \alpha} } & \\ \tan \dfrac{\alpha }{2}= \dfrac{\sin \dfrac{\alpha }{2}}{\cos \dfrac{\alpha }{2}} = \dfrac{2\sin \dfrac{\alpha }{2}\cos \dfrac{\alpha }{2}}{2\cos^2 \dfrac{\alpha }{2}} = \dfrac{\sin \alpha }{1+\cos \alpha } & \\ \tan \dfrac{\alpha }{2}= \dfrac{\sin \dfrac{\alpha }{2}}{\cos \dfrac{\alpha }{2}} = \dfrac{2\sin^2 \dfrac{\alpha }{2}}{2\sin \dfrac{\alpha }{2}\cos \dfrac{\alpha }{2}} =\dfrac{1-\cos \alpha }{\sin \alpha } \\ \end{array}

Triple angle identities

\def\arraystretch{1.5} \begin{array}{ll} \sin 3\alpha =3\sin \alpha -4\sin^3\alpha &\\ \cos 3\alpha =4\cos^3\alpha -3\cos \alpha &\\ \tan 3\alpha =\dfrac{3\tan \alpha -\tan^3\alpha }{1-3\tan^2\alpha } &\\ \end{array}

Triple identities transform a trig function of an angle to a cubic equation in terms of trigonometric function of an angle that is one-third of its value.

Trigonometric Values of Special Angles

With the help of trigonometric identities and special triangle, we can derive the exact value of the special angle. Since these functions are periodic functions, we only need consider the angle range from 1\degree to 90\degree. Moreover, trig values of the half of the angles are equal to the values of the cofunction of the complementary angles.

What are the Special Angles for Trigonometric Functions

Special angles are those whose trigonometric functions are could be represented by expressions of real numbers. These angles are normally multiples of 3. For those not multiples of 3, normally they have results of 2 complex number. Refer to the article Exact Trigonometric Values from 1° to 90° for more details.

Table for Exact Values of Basic Special Angles

Trigonometric values of 0, 30\degree, 45\degree, 60\degree, 90\degree are based on two special right-angles triangles 30-60-90 and 45-45-90 as well as their degenerate cases of a horizontal or vertical line. Trigonometric values of other angles of multiples of 3 could be computed by converting them as expressions of trig functions of these angles. Results are given in the following tables.

\def\arraystretch{2.4} \begin{array}{|c|c|c|c|c|c|} \hline &0\degree &30\degree&45\degree&60\degree&90\degree \\ \hline \sin&0&\dfrac{1}{2}&\dfrac{\sqrt{2} }{2}&\dfrac{\sqrt{3} }{2}&1 \\ \hline \cos&1&\dfrac{\sqrt{3} }{2}&\dfrac{\sqrt{2} }{2}&\dfrac{1}{2} &0\\ \hline \tan&0&\dfrac{\sqrt{3} }{3}&1&\sqrt{3} &\infty\\ \hline \csc&\infty&2&\sqrt{2}&\dfrac{2\sqrt{3} }{3} &1\\ \hline \sec&1&\dfrac{2\sqrt{3} }{3}&\sqrt{2} &2&\infty \\ \hline \cot&\infty&\sqrt{3}&1&\dfrac{\sqrt{3} }{3} &0\\ \hline \end{array}

Table for Exact Trigonometric Values of 15°

sin 15° could be determined by using the difference identity for sines function or half angle identities. cos 15° could be determined by using the difference identity for cosines function or half angle identities.

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline & 15° \\ \hline \sin& \dfrac{\sqrt{6}-\sqrt{2} }{4} \\ \hline \cos& \dfrac{\sqrt{6}+\sqrt{2} }{4} \\ \hline \tan& \dfrac{\sqrt{6}-\sqrt{2} }{\sqrt{6}+\sqrt{2} } =2-\sqrt{3} \\ \hline \csc& \dfrac{4}{\sqrt{6}-\sqrt{2} } =\sqrt{6}+\sqrt{2} \\ \hline \sec&\dfrac{4}{\sqrt{6}+\sqrt{2}} =\sqrt{6}-\sqrt{2} \\ \hline \cot&\dfrac{1}{2-\sqrt{3} } = 2+\sqrt{3} \\ \hline \end{array}

Table for Exact Trigonometric Values of 18°

Derivation of exact trig values of 18° involves using of a golden triangle or using triple identities to determine the exact value of sin 18° and cos 18° .

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline & 18° \\ \hline \sin& \dfrac{-1+\sqrt{5} }{4} \\ \hline \cos& \dfrac{\sqrt{10+2\sqrt{5} } }{4} \\ \hline \tan&\dfrac{-1+\sqrt{5} }{\sqrt{10+2\sqrt{5} }} \\ \hline \csc&\dfrac{4}{\sqrt{5} -1 } =\sqrt{5} +1 \\ \hline \sec&\dfrac{4}{\sqrt{10+2\sqrt{5} } } =\dfrac{2\sqrt{5} }{5}(2-\sqrt{2} ) \\ \hline \cot&\dfrac{\sqrt{10+2\sqrt{5} }}{-1+\sqrt{5} } \\ \hline \end{array}

Table for Exact Trigonometric Values of 12°

Sin 12° and cos 12° could be determined by using the difference identities.

\sin 12° = \sin(72° -60° ) = \sin 72°\cos 60° -\cos 72° \sin 60°

\cos 12° = \cos(72° -60° ) = \cos 72°\cos 60° +\sin 72° \sin 60°

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline &12° \\ \hline \sin& \dfrac{1}{8}[ \sqrt{10+2\sqrt{5} } -\sqrt{3}( -1+\sqrt{5} ) ] \\ \hline \cos& \dfrac{1}{8}[ (-1+\sqrt{5} )+\sqrt{3}\sqrt{10+2\sqrt{5} } ] \\ \hline \tan&\dfrac{ \sqrt{10+2\sqrt{5} } -\sqrt{3}( -1+\sqrt{5} )}{ (-1+\sqrt{5} )+\sqrt{3}\sqrt{10+2\sqrt{5} }} \\ \hline \csc&\dfrac{8}{\sqrt{10+2\sqrt{5} } -\sqrt{3}( -1+\sqrt{5} ) } \\ \hline \sec&\dfrac{8}{ (-1+\sqrt{5} )+\sqrt{3}\sqrt{10+2\sqrt{5} }} \\ \hline \cot& \dfrac{ (-1+\sqrt{5} )+\sqrt{3}\sqrt{10+2\sqrt{5} }}{\sqrt{10+2\sqrt{5} } -\sqrt{3}( -1+\sqrt{5} ) } \\ \hline \end{array}

Table for Exact Trigonometric Values of

Trig values of 9° could be transformed to an expression in terms of trig functions of 45° and 36°.

\sin 9° = \sin(45° -36° ) = \sin 45°\cos 36° -\cos 45° \sin 36°

\cos 9° = \cos(45° -36° ) = \cos 45°\cos 36° +\sin 45° \sin 36°

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline & 9° \\ \hline \sin& \dfrac{1}{8}[ (\sqrt{2} +\sqrt{10}) - \sqrt{20-4\sqrt{5} }] \\ \hline \cos& \dfrac{1}{8}[(\sqrt{2} +\sqrt{10}) + \sqrt{20-4\sqrt{5} } ] \\ \hline \tan& \dfrac{ (\sqrt{2} +\sqrt{10}) - \sqrt{20-4\sqrt{5} }}{(\sqrt{2} +\sqrt{10}) + \sqrt{20-4\sqrt{5} }} \\ \hline \csc&\dfrac{8}{(\sqrt{2} +\sqrt{10}) - \sqrt{20-4\sqrt{5} }} \\ \hline \sec&\dfrac{8}{(\sqrt{2} +\sqrt{10}) + \sqrt{20-4\sqrt{5} } } \\ \hline \cot& \dfrac{(\sqrt{2} +\sqrt{10}) + \sqrt{20-4\sqrt{5} } }{(\sqrt{2} +\sqrt{10}) - \sqrt{20-4\sqrt{5} }} \\ \hline \end{array}

Table for Exact Trigonometric Values of

Trig values of 6° could be transformed to an expression in terms of sine and cosine of 36° and 30°.

\sin 6° = \sin(36° -30° ) = \sin 36°\cos 30° -\cos 36° \sin 30°

\cos 6° = \cos(36° -30° ) = \cos 36°\cos 30° +\sin 36° \sin 30°

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline & 6° \\ \hline \sin& \dfrac{1}{8}[\sqrt{3} \sqrt{10-2\sqrt{5} } - \sqrt{10-2\sqrt{5} } ] \\ \hline \cos& \dfrac{1}{8}[\sqrt{3}(1+\sqrt{5} ) + \sqrt{10-2\sqrt{5} } ] \\ \hline \tan&\dfrac{\sqrt{3} \sqrt{10-2\sqrt{5} } - \sqrt{10-2\sqrt{5} } }{\sqrt{3}(1+\sqrt{5} ) + \sqrt{10-2\sqrt{5} } } \\ \hline \csc&\dfrac{8}{\sqrt{3} \sqrt{10-2\sqrt{5} } - \sqrt{10-2\sqrt{5} }} \\ \hline \sec&\dfrac{8}{\sqrt{3}(1+\sqrt{5} ) + \sqrt{10-2\sqrt{5} } } \\ \hline \cot&\dfrac{\sqrt{3}(1+\sqrt{5} ) + \sqrt{10-2\sqrt{5} } }{\sqrt{3} \sqrt{10-2\sqrt{5} } - \sqrt{10-2\sqrt{5} } } \\ \hline \end{array}

Collected in the board: Trigonometry

Steven Zheng Steven Zheng posted 1 month ago

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