Wrapping Up Trigonometry

Solving Trigonometry Equations

To solve trig equations, first isolate the trig function of sinx, tanx, etc. Then, just substitute values that make the equation true. Make sure to find all solutions under the domain.

Law of Sines

Ex: If triangle ABC has side length AB=10 and angles B=57° and C=49°, what is side length AC? sinB/b = sinC/c sin57°/b = sin49°/10 0.839/b = 0.755/10 0.839/b = 0.075 0.839 = 0.075b 9.683 = b

Solving Trigonometry Equations

Ex: sinx+1=-sinx for values in the domain [0, 2pi]. Combine like terms. 2sinx=-1. Divide. sinx=-1/2. The values of sinx that yield -1/2 and are in the range are 7pi/6 and 11pi/6, so x = 7pi/6, 11pi/6. Ex: 4cos^2(x)-3=0 for values in the domain [0, 2pi]. Isolate cos^2(x) first. cos^2(x)=3/4. Square root both sides to isolate cosx. cosx=(sqrt3)/2. The values of cosx that make this true are pi/6 and 11pi/6, so x=pi/6, 11pi/6.

Law of Sines

Can also be written as sinA/a = sinB/b = sinC/c

Law of Cosines

Can substitute any side for c.

Converting between Degrees/Radians

Ex: Convert 5pi/6 to degrees. 5pi/*180°/pi=150°. Ex: Convert 144° to radians. 144°*pi/180°=4pi/5.

Evaluating Inverse Trigonometric Functions

Ex: Find arccos(1/2). The angle that yields cosx=1/2 and is in the range [0, pi] is pi/3. Thus arccos(1/2)=pi/3. Ex: Find arcsin(sin(pi)). sin(pi) is simply 0, so the expression becomes arcsin(0). The angle that yields sinx=0 and is the range [-pi/2, pi/2] is 0. Thus, arsin(sin(pi))=0

Evaluating Inverse Trigonometric Functions

The inverse, or arctrig functions effectively cancel out the normal trig function. For example, sin(pi/6)=1/2, but arcsin(1/2)=pi/6. However these functions also have range restrictions. For arcsinx, it is [-pi/2, pi/2]. For arccosx, it is [0, pi]. For arctanx, it is (-pi/2, pi/2).

Converting between Degrees/Radians

There are 360° in a full rotation. There are 2pi radians in a full rotation. To convert from degrees to radians, multiply by pi/180°. To convert from radians to degrees, multiply by 180°/pi.

Verifying Identities

To verify more complex trig identities, use simpler identities like tanx=sinx/cosx, cotx=cosx/sinx, secx=1/cosx, cscx=1/sinx, and sin^2(x)+cos^2(x)=1.

Evaluating Six Trigonometry Functions at Unit Circle Points

Ex: Find sin(pi/3). On the unit circle, pi/3 corresponds to the point (1/2, (sqrt3)/2). Since we are trying to find sin(pi/3), we take the y-coordinate of (sqrt3)/2. Ex: Find tan(pi/4). On the unit circle, pi/4 corresponds to the point ((sqrt2)/2, (sqrt2)/2). Since we are trying to find tan(pi/4), we divide the y-coordinate by the x-coordinate and get (sqrt2)/2 / (sqrt2)/2=1.

Law of Cosines

Ex: If triangle ABC has side lengths AB=4, BC=5, and AC=6, what is angle C? c^2 = a^2 + b^2 - 2ab cosC 4^2 = 5^2 + 6^2 - 2(5)(6) cosC 16 = 61 - 60 cosC -45 = -60cosC 3/4 = cosC 41.410 = C

Verifying Identities

Ex: Prove that sinx(cotx+tanx)=secx. First we can write in terms of sinx and cosx. sinx(cosx/sinx + sinx/cosx)=secx. Get a common demoninator. sinx((cos^2(x)+sin^2(x))/sinxcosx)=secx. Apply Pythagorean Identity. sinx(1/sinxcosx)=secx. Cross out like terms. 1/cosx=secx. This is just a reciprocal identity and we end up with secx=secx, thus proving the statement.

Evaluating Six Trigonometry Functions at Unit Circle Points

To find sinx, take the y-coordinate of the ordered pair that corresponds to the angle on the unit circle. To find cosx, take the x-coordinate. To find tanx, divide the y-coordinate by the x-coordinate. To find cscx, take the reciprocal of sinx. To find secx, take the reciprocal of cosx. To find cotx, take the reciprocal of tanx .