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A confusing topic for some with its abstract functions and formulas, trigonometry can leave many students lost and confused, but it doesn’t have to be a mystery once you understand its fundamental formulas and identities.
This post has formulas and identities needed for trigonometry. Near the bottom, there is a helpful app listed where you can learn essential trigonometric formulas and apply them in practice problems efficiently.
Table of Contents
Trigonometry Formulas
• Right Triangle Definition
$$sin\theta=\frac{Opposite}{Hypotenuse} \hspace{2cm} cos\theta=\frac{Adjacent}{Hypotenuse}$$
$$tan\theta=\frac{Opposite}{Adjacent} \hspace{2.5cm} sec\theta=\frac{Hypotenuse}{Adjacent}$$
$$csc\theta=\frac{Hypotenuse}{Opposite} \hspace{2cm} cot\theta=\frac{Adjacent}{Opposite} \hspace{0.6cm}$$
• Reciprocal Identities
$$sin\theta=\frac{1}{csc\theta} \hspace{2cm} cos\theta=\frac{1}{sec\theta}$$
$$tan\theta=\frac{1}{cot\theta} \hspace{2cm} csc\theta=\frac{1}{sin\theta}$$
$$sec\theta=\frac{1}{cos\theta} \hspace{2cm} cot\theta=\frac{1}{tan\theta}$$
• Quotient Identities
$$tan\theta=\frac{sin\theta}{cos\theta} \hspace{2cm} cot\theta=\frac{cos\theta}{sin\theta}$$
• Special Angles
• Sign of Trigonometric Functions
• Unit Circle
$$sin\theta=\frac{y}{1} \hspace{2cm} csc\theta=\frac{1}{y}$$
$$cos\theta=\frac{x}{1} \hspace{2cm} sec\theta=\frac{1}{x}$$
$$tan\theta=\frac{y}{x} \hspace{2cm} cot\theta=\frac{x}{y}$$
• Cofunction Identities
$$sin(\frac{\pi}{2}-\theta)=cos\theta \hspace{2cm} cos(\frac{\pi}{2}-\theta)=sin\theta$$
$$tan(\frac{\pi}{2}-\theta)=cot\theta \hspace{2cm} sec(\frac{\pi}{2}-\theta)=csc\theta$$
$$csc(\frac{\pi}{2}-\theta)=sec\theta \hspace{2cm} cot(\frac{\pi}{2}-\theta)=tan\theta$$
• Supplementary Angles Identities
$$sin(\pi-\theta)=sin\theta \hspace{2cm} cos(\pi-\theta)=-cos\theta$$
$$tan(\pi-\theta)=-tan\theta \hspace{1.55cm} csc(\pi-\theta)=csc\theta \hspace{0.5cm}$$
$$sec(\pi-\theta)=-sec\theta \hspace{1.75cm} cot(\pi-\theta)=-cot\theta \hspace{0.2cm}$$
• Periodicity Identities
Sin, cos, csc, and sec have a period of 2π. Tan and cot have a period of π.
$$sin(2n\pi+\theta)=sin\theta \hspace{2cm} cos(2n\pi+\theta)=cos\theta$$
$$csc(2n\pi+\theta)=csc\theta \hspace{2cm} sec(2n\pi+\theta)=sec\theta$$
$$tan(n\pi+\theta)=tan\theta \hspace{2.2cm} cot(n\pi+\theta)=cot\theta \hspace{0.35cm}$$
• Odd-Even Identities
$$sin(-θ)=-sinθ \hspace{2cm} cos(-θ)=cosθ \hspace{0.2cm}$$
$$tan(-θ)=-tanθ \hspace{1.95cm} cot(-θ)=-cotθ$$
$$sec(-θ)=secθ \hspace{2.4cm} csc(-θ)=-cscθ$$
• Sum and Difference Identities
$$sin(α+β)=sinα⋅cosβ+cosα⋅sinβ$$
$$cos(α+β)=cosα⋅cosβ-sinα⋅sinβ$$
$$sin(α-β)=sinα⋅cosβ-cosα⋅sinβ$$
$$cos(α – β)=cosα⋅cosβ+sinα⋅sinβ$$
$$tan(α+β)=\frac{tanα+tanβ}{1-tanα⋅tanβ}$$
$$tan(α-β)=\frac{tanα-tanβ}{1+tanα⋅tanβ}$$
• Half-Angle Identities
$$sin(\frac{θ}{2})=±\sqrt{\frac{1-cosθ}{2}}$$
$$cos(\frac{θ}{2})=±\sqrt{\frac{1+cosθ}{2}}$$
$$tan(\frac{θ}{2})=\frac{1-cosθ}{sinθ}=\frac{sinθ}{1+cosθ}=±\sqrt{\frac{1-cosθ}{1+cosθ}}$$
• Double Angle Identities
$$sin(2θ)=2sinθ⋅cosθ=\frac{2tanθ}{1+tan^{2}θ}$$
$$cos(2θ)=cos^{2}θ-sin^{2}θ=2cos^{2}θ-1=1-2sin^{2}θ=\frac{1-tan^{2}θ}{1+tan^{2}θ}$$
$$tan(2θ)=\frac{2tanθ}{1-tan^{2}θ}$$
$$sec(2θ)=\frac{sec^{2}θ}{2-sec^{2}θ}$$
$$csc(2θ)=\frac{secθ⋅cscθ}{2}$$
• Triple Angle Identities
$$sin(3θ)=3sinθ-4sin^{3}θ$$
$$cos(3θ)=4cos^{3}θ-3cosθ$$
$$tan(3θ)=\frac{3tanθ-tan^{3}θ}{1-3tan^{2}θ}$$
• Product to Sum Formulas
$$sinα⋅cosβ=\frac{sin(α+β)+sin(α-β)}{2}$$
$$cosα⋅sinβ=\frac{sin(α+β)-sin(α-β)}{2}$$
$$cosα⋅cosβ=\frac{cos(α+β)+cos(α-β)}{2}$$
$$sinα⋅sinβ=\frac{cos(α-β)-cos(α+β)}{2}$$
• Sum to Product Formulas
$$sinα+sinβ=2sin(\frac{α+β}{2})⋅cos(\frac{α-β}{2})$$
$$sinα-sinβ=2sin(\frac{α-β}{2})⋅cos(\frac{α+β}{2})$$
$$cosα+cosβ=2cos(\frac{α+β}{2})⋅cos(\frac{α-β}{2})$$
$$cosα-cosβ=-2sin(\frac{α+β}{2})⋅sin(\frac{α-β}{2})$$
• Power Reducing Formulas
$$sin^{2}θ=\frac{1-cos(2θ)}{2} \hspace{2cm} cos^{2}θ=\frac{1+cos(2θ)}{2}$$
$$tan^{2}θ=\frac{1-cos(2θ)}{1+cos(2θ)} \hspace{2cm} csc^{2}θ=\frac{2}{1-cos(2θ)}$$
$$cot^{2}θ=\frac{1+cos(2θ)}{1-cos(2θ)} \hspace{2cm} sec^{2}θ=\frac{2}{1+cos(2θ)}$$
• Inverse Trigonometry Formulas
$$sin^{-1}(-x)=-sin^{-1}x \hspace{2.4cm} cos^{-1}(-x)=π-cos^{-1}x$$
$$tan^{-1}(-x)=-tan^{-1}x \hspace{2.3cm} csc^{-1}(-x)=-csc^{-1}x \hspace{0.5cm}$$
$$sec^{-1}(-x)=π-sec^{-1}x \hspace{2cm} cot^{-1}(-x)=π-cot^{-1}x \hspace{0.1cm}$$
• Pythagorean Identities
$$sin^{2}θ+cos^{2}θ=1$$
$$sec^{2}θ-tan^{2}θ=1$$
$$csc^{2}θ-cot^{2}θ=1$$
• Law of Sine
$$\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$$
• Law of Cosine
$$a^{2}=b^{2}+c^{2}-2bc⋅cosA$$
$$b^{2}=a^{2}+c^{2}-2ac⋅cosB$$
$$c^{2}=a^{2}+b^{2}-2ab⋅cosC$$
• Side Angle Side Formula
$$Area=\frac{1}{2}ab⋅sinC$$
Trigonometry Study App
This app explores a wide range of essential trigonometry formulas, including sine, cosine, tangent, cofunction, periodicity, and advanced identities such as Pythagorean, double angle, sum and difference identities, and sum-to-product formulas.
The app keeps your study sessions efficient by using the built-in reference for quick access to formulas and graphs. The app contains an array of questions with two quiz levels. It is perfect for students or anyone looking to strengthen their trigonometry skills.
The app can be downloaded from the Google Play Store.
