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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 high school/college students or anyone looking to strengthen their trigonometry skills.
The app can be downloaded from the Google Play Store.
