Trigonometric Identities and Equations

Are these expressions equivalent? Why or why not?

a.  $${\mathrm{sin}(a+b)}$$          b.  $${\mathrm{sin}a+\mathrm{sin}b}$$

How can you justify your reasoning when $${a={\pi\over6}}$$ and $${b={\pi\over3}}$$? Use the unit circle shown below to justify your thinking.

Below are the sum and difference formulas for sine, cosine, and tangent.

$${\mathrm{sin}(a+b)=\mathrm{sin}a\mathrm{cos}b+\mathrm{cos}a\mathrm{sin}b}$$

$${\mathrm{sin}(a-b)=\mathrm{sin}a\mathrm{cos}b-\mathrm{cos}a\mathrm{sin}b}$$

$${\mathrm{cos}(a+b)=\mathrm{cos}a\mathrm{cos}b-\mathrm{sin}a\mathrm{sin}b}$$

$${\mathrm{cos}(a-b)=\mathrm{cos}a\mathrm{cos}b+\mathrm{sin}a\mathrm{sin}b}$$

$${\mathrm{tan}(a+b)={\mathrm{tan}a+\mathrm{tan}b\over{1-\mathrm{tan}a\mathrm{tan}b}}}$$

$${\mathrm{tan}(a-b)={\mathrm{tan}a-\mathrm{tan}b\over{1+\mathrm{tan}a\mathrm{tan}b}}}$$

Use both sum and difference formulas to find the value of:

$${\mathrm{sin}(75^{\circ})}$$

$${\mathrm{tan}(15^{\circ})}$$

$${\mathrm{cos}(105^{\circ})}$$

Find $${\mathrm{cos}(a+b)}$$ given that $${\mathrm{cos}(a)=-{4\over5}}$$ with $${\pi \leq a \leq {3\pi\over2}}$$ and $${\mathrm{sin}(b)={5\over13}}$$ with $${0 \leq b \leq {\pi\over2}}$$.

Find $${\mathrm{sin}(u-v)}$$ given $${\mathrm{sin}(u)={8\over17}}$$ with $${0\leq u \leq {\pi\over2}}$$ and $${\mathrm{cos}(v)=-{24\over25}}$$ with $${\pi \leq v \leq {3\pi\over2}}$$.

Evaluate: 

$${\mathrm{sin}(-15^{\circ})}$$

$${\mathrm{cos}(15^{\circ})}$$