Test Objectives
  • Demonstrate the ability to use the fundamental identities to find function values
  • Demonstrate the ability to use the fundamental identities to perform transformations
Fundamental Identities Practice Test:

#1:

Instructions: Use the fundamental identities to find the missing trig ratios.

$$a)\hspace{.1em}\text{cot}\hspace{.15em}θ=-\frac{15}{8}, \text{sin}\hspace{.15em}θ < 0$$

$$b)\hspace{.1em}\text{csc}\hspace{.15em}θ=-\frac{5}{3}, \text{cos}\hspace{.15em}θ > 0$$


#2:

Instructions: Use the fundamental identities to find the missing trig ratios.

$$a)\hspace{.1em}\text{tan}\hspace{.15em}θ=\frac{3}{4}, \text{cos}\hspace{.15em}θ < 0$$

$$b)\hspace{.1em}\text{sin}\hspace{.15em}θ=\frac{7}{11}, \text{cos}\hspace{.15em}θ > 0$$


#3:

Instructions: Use the fundamental identities to find the missing trig ratios.

$$a)\hspace{.1em}\text{cot}\hspace{.15em}θ=2\sqrt{2}, \text{cos}\hspace{.15em}θ > 0$$

$$b)\hspace{.1em}\text{sec}\hspace{.15em}θ=\frac{\sqrt{10}}{3}, \text{sin}\hspace{.15em}θ < 0$$


#4:

Instructions: Perform the indicated transformation.

$$a)\hspace{.1em}\text{Write sin θ in terms of cos θ}$$

$$b)\hspace{.1em}\text{Write tan θ in terms of sec θ}$$


#5:

Instructions: Write in terms of sine and cosine, remove all quotients.

$$a)\hspace{.1em}\text{sin}^2 \hspace{.1em}θ(\text{csc}^2 \hspace{.1em}θ - 1)$$

$$b) \hspace{.1em}\frac{1 - \text{cos}^2(-θ)}{1 + \text{tan}^2(-θ)}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}$$ $$\text{sin}\hspace{.1em}θ=-\frac{8}{17}$$ $$\text{cos}\hspace{.1em}θ=\frac{15}{17}$$ $$\text{tan}\hspace{.1em}θ=-\frac{8}{15}$$ $$\text{sec}\hspace{.1em}θ=\frac{17}{15}$$ $$\text{csc}\hspace{.1em}θ=-\frac{17}{8}$$

$$b)\hspace{.1em}$$ $$\text{sin}\hspace{.1em}θ=-\frac{3}{5}$$ $$\text{cos}\hspace{.1em}θ=\frac{4}{5}$$ $$\text{tan}\hspace{.1em}θ=-\frac{3}{4}$$ $$\text{sec}\hspace{.1em}θ=\frac{5}{4}$$ $$\text{cot}\hspace{.1em}θ=-\frac{4}{3}$$


#2:

Solutions:

$$a)\hspace{.1em}$$ $$\text{sin}\hspace{.1em}θ=-\frac{3}{5}$$ $$\text{cos}\hspace{.1em}θ=-\frac{4}{5}$$ $$\text{sec}\hspace{.1em}θ=-\frac{5}{4}$$ $$\text{csc}\hspace{.1em}θ=-\frac{5}{3}$$ $$\text{cot}\hspace{.1em}θ=\frac{4}{3}$$

$$b)\hspace{.1em}$$ $$\text{cos}\hspace{.1em}θ=\frac{6 \sqrt{2}}{11}$$ $$\text{tan}\hspace{.1em}θ=\frac{7 \sqrt{2}}{12}$$ $$\text{sec}\hspace{.1em}θ=\frac{11 \sqrt{2}}{12}$$ $$\text{csc}\hspace{.1em}θ=\frac{11}{7}$$ $$\text{cot}\hspace{.1em}θ=\frac{6 \sqrt{2}}{7}$$


#3:

Solutions:

$$a)\hspace{.1em}$$ $$\text{sin}\hspace{.1em}θ=\frac{1}{3}$$ $$\text{cos}\hspace{.1em}θ=\frac{2 \sqrt{2}}{3}$$ $$\text{tan}\hspace{.1em}θ=\frac{\sqrt2}{4}$$ $$\text{sec}\hspace{.1em}θ=\frac{3 \sqrt{2}}{4}$$ $$\text{csc}\hspace{.1em}θ=3$$

$$b)\hspace{.1em}$$ $$\text{sin}\hspace{.1em}θ=-\frac{\sqrt{10}}{10}$$ $$\text{cos}\hspace{.1em}θ=\frac{3 \sqrt{10}}{10}$$ $$\text{tan}\hspace{.1em}θ=-\frac{1}{3}$$ $$\text{csc}\hspace{.1em}θ=-\sqrt{10}$$ $$\text{cot}\hspace{.1em}θ=-3$$


#4:

Solutions:

$$a)\hspace{.1em}\text{sin θ}=\pm \sqrt{1 - \text{cos}^2 \hspace{.15em}θ}$$

$$b)\hspace{.1em}\text{tan θ}=\pm \sqrt{\text{sec}^2 \hspace{.15em}θ - 1}$$


#5:

Solutions:

$$a)\hspace{.1em}\text{cos}^2 \hspace{.15em}θ$$

$$b)\hspace{.1em}\text{sin}^2 \hspace{.15em}θ \hspace{.1em}\text{cos}^2 \hspace{.15em}θ$$