Functions

Special Triangles and Trigonometric Identities

Special Triangles

\displaystyle 45^\circ,\ 45^\circ,\ 90^\circ

\displaystyle 30^\circ,\ 60^\circ,\ 90^\circ

QUESTION

Which is the value of

\displaystyle \cos\left(\dfrac\pi4\right)

Trigonometric Identities

THEOREM

Trigonometric Identities

\begin{aligned} & \sin^2(\theta) + \cos^2(\theta) = 1 \\[0.65em] & \sin(-\theta) = -\sin(\theta) \\[0.65em] & \cos(-\theta) = \cos(\theta) \\[0.65em] & \tan(\theta) = {\dfrac{\sin(\theta)}{\cos(\theta)}} \\[0.65em] & \cot(\theta) = {\dfrac{\cos(\theta)}{\sin(\theta)}} \\[0.65em] & \tan^2(\theta) + 1 = \sec^2(\theta) \\[0.65em] & \cot^2(\theta) + 1 = \csc^2(\theta) \\[0.65em] & \sin(\theta) = \cos\left(\theta - {\dfrac{\pi}{2}}\right) \\[0.65em] & \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \\[0.65em] & \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \\[0.65em] & \sin(2\theta) = 2\sin(\theta)\cos(\theta) \\[0.65em] & \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \\[0.65em] & \sin^2(\theta) = {\dfrac{1 - \cos(2\theta)}{2}} \\[0.65em] & \cos^2(\theta) = {\dfrac{1 + \cos(2\theta)}{2}} \end{aligned}

Reminder: $trig^n(x)$ is a notation often used to indicate $(trig(x))^n$

These identities hold wherever both sides are defined.

QUESTION

Which statement is true for every real

\displaystyle \theta

Evaluate each of the following.

(a) $\displaystyle \sin(\pi/4)$

(b) $\displaystyle \cos(\pi/4)$

Find all values of $\displaystyle \theta$ satisfying the following.

(a) $\displaystyle \tan(\theta)=1/\sqrt{3}$

(b) $\displaystyle \sec(\theta)=\sqrt{2}$

◀ Previous SectionTrigonometric Functions Next Section ▶Exponential Functions

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Functions

Special Triangles and Trigonometric Identities

Special Triangles

\displaystyle 45^\circ,\ 45^\circ,\ 90^\circ

\displaystyle 30^\circ,\ 60^\circ,\ 90^\circ

QUESTION

Which is the value of

\displaystyle \cos\left(\dfrac\pi4\right)

Trigonometric Identities

THEOREM

Trigonometric Identities

\begin{aligned} & \sin^2(\theta) + \cos^2(\theta) = 1 \\[0.65em] & \sin(-\theta) = -\sin(\theta) \\[0.65em] & \cos(-\theta) = \cos(\theta) \\[0.65em] & \tan(\theta) = {\dfrac{\sin(\theta)}{\cos(\theta)}} \\[0.65em] & \cot(\theta) = {\dfrac{\cos(\theta)}{\sin(\theta)}} \\[0.65em] & \tan^2(\theta) + 1 = \sec^2(\theta) \\[0.65em] & \cot^2(\theta) + 1 = \csc^2(\theta) \\[0.65em] & \sin(\theta) = \cos\left(\theta - {\dfrac{\pi}{2}}\right) \\[0.65em] & \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \\[0.65em] & \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \\[0.65em] & \sin(2\theta) = 2\sin(\theta)\cos(\theta) \\[0.65em] & \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \\[0.65em] & \sin^2(\theta) = {\dfrac{1 - \cos(2\theta)}{2}} \\[0.65em] & \cos^2(\theta) = {\dfrac{1 + \cos(2\theta)}{2}} \end{aligned}

Reminder: $trig^n(x)$ is a notation often used to indicate $(trig(x))^n$

These identities hold wherever both sides are defined.

QUESTION

Which statement is true for every real

\displaystyle \theta

Evaluate each of the following.

(a) $\displaystyle \sin(\pi/4)$

(b) $\displaystyle \cos(\pi/4)$

Find all values of $\displaystyle \theta$ satisfying the following.

(a) $\displaystyle \tan(\theta)=1/\sqrt{3}$

(b) $\displaystyle \sec(\theta)=\sqrt{2}$

◀ Previous SectionTrigonometric Functions Next Section ▶Exponential Functions

Special Triangles and Trigonometric Identities

Special Triangles

KNOWLEDGE QUIZ

Trigonometric Identities

Trigonometric Identities

KNOWLEDGE QUIZ

Special trigonometric values

All solutions of trigonometric equations

Special Triangles and Trigonometric Identities

Special Triangles

KNOWLEDGE QUIZ

Trigonometric Identities

Trigonometric Identities

KNOWLEDGE QUIZ

Special trigonometric values

All solutions of trigonometric equations