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Verify the identity: tan(π−θ)=−tan θ.
Let L1 and L2 denote two non-vertical intersecting lines, and let θ denote the acute angle between L1 and L2. See [link] . Show that
where m1 and m2 are the slopes of L1 and L2 respectively. ( Hint: Use the fact that tan θ1=m1 and tan θ2=m2. )
Using the difference formula for tangent, this problem does not seem as daunting as it might.
For a climbing wall, a guy-wire R is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire S attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle α between the wires. See [link] .
Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that tan β=4750, and tan(β−α)=4050=45. We can then use difference formula for tangent.
Now, substituting the values we know into the formula, we have
Use the distributive property, and then simplify the functions.
Now we can calculate the angle in degrees.
Access these online resources for additional instruction and practice with sum and difference identities.
Sum Formula for Cosine | cos(α+β)=cos α cos β−sin αsin β |
Difference Formula for Cosine | cos(α−β)=cos α cos β+sin α sin β |
Sum Formula for Sine | sin(α+β)=sin α cos β+cos α sin β |
Difference Formula for Sine | sin(α−β)=sin α cos β−cos α sin β |
Sum Formula for Tangent | tan(α+β)=tan α+tan β1−tan α tan β |
Difference Formula for Tangent | tan(α−β)=tan α−tan β1+tan α tan β |
Cofunction identities | sin θ=cos(π2−θ)cos θ=sin(π2−θ)tan θ=cot(π2−θ)cot θ=tan(π2−θ)sec θ=csc(π2−θ)csc θ=sec(π2−θ) |
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