Math III
Go further with polynomial and rational expressions, advanced functions, trigonometry, geometric modeling, and statistical inference.
- Problem types
- 641
- Practice variants
- 2,564
Page 15 of 18
Each problem type has four distinct practice variants. Open a preview to move among all four.
Solve a right-triangle design dimension to stated precision
Use geometric methods to solve design problems under constraints such as cost, space, or ratios.
Fix the right-triangle geometry before calculating: label the known side and unknown side relative to the given angle. Use the trigonometric ratio containing exactly those two roles, solve the equation …
Preview problemUse the Law of Sines to find a missing side from two angles and one opposite side
Prove the Laws of Sines and Cosines and use them to solve problems.
Build the Law of Sines proportion from opposite pairs, not from where labels happen to sit in the drawing. Anchor the proportion with the fully known side-angle pair and match …
Preview problemFind all valid SSA angles and triangle count
Prove the Laws of Sines and Cosines and use them to solve problems.
An SSA setup needs an ambiguity audit after inverse sine. Compute the principal angle, form its supplement, and test each candidate with the given angle; keep a candidate only if …
Preview problemUse the Law of Cosines to find a missing side from two sides and the included angle
Prove the Laws of Sines and Cosines and use them to solve problems.
With two sides and their included angle, match the unknown side to the angle opposite it and use the corresponding Law of Cosines form. Evaluate the cosine correction carefully, then …
Preview problemFind a Law-of-Cosines angle to stated precision
Prove the Laws of Sines and Cosines and use them to solve problems.
When all three sides are known, first identify which side is opposite the requested angle. Rearrange the Law of Cosines to isolate that angle's cosine, simplify the exact ratio, and …
Preview problemChoose whether to use the Law of Sines or the Law of Cosines first
Prove the Laws of Sines and Cosines and use them to solve problems.
Inventory the given triangle information before deciding on a law. A known side with its opposite angle creates the proportion structure used by the Law of Sines, while three sides …
Preview problemClassify acute SSA with altitude and list all angles
Prove the Laws of Sines and Cosines and use them to solve problems.
For an acute SSA case, use the given adjacent side and angle to compute the altitude before solving any angles. Comparing the opposite side with that altitude and the adjacent …
Preview problemSolve and verify all six triangle measures
Prove the Laws of Sines and Cosines and use them to solve problems.
Complete the angle sum first so every opposite side-angle pair is available. Use the fully known pair to anchor one Law of Sines scale, then apply that same scale to …
Preview problemSelect a triangle law and complete the measurement
Prove the Laws of Sines and Cosines and use them to solve problems.
Translate the two measured rays and the angle between them into an SAS triangle. The desired separation lies opposite the included angle, so match it to the opposite-side form of …
Preview problemClassify and interpret a solved triangle result
Prove the Laws of Sines and Cosines and use them to solve problems.
Interpret a solved value by naming the physical quantity and carrying its units into the conclusion. Then test the restrictions relevant to that quantity and triangle: lengths must be positive, …
Preview problemSolve both unknown sides in an AAS/ASA triangle
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
In an ASA or AAS triangle, complete the third angle before setting up side proportions. Match every lowercase side with its opposite capital angle, and use the fully known pair …
Preview problemReport every complete SSA triangle solution
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
An SSA solution is not complete after the calculator returns one inverse-sine angle. Test that angle and its supplement separately, rejecting any candidate that leaves a nonpositive third angle. For …
Preview problemUse the Law of Cosines to find a missing side when two sides and the included angle are given
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Recognize the two known sides and the angle between them as an SAS pattern. The unknown closing side must be paired with that included angle because they are opposite each …
Preview problemCalculate and classify an SSS triangle's largest angle
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
The largest angle is always opposite the longest side, so target that pair first in an SSS triangle. Isolate the angle's cosine using the other two sides as adjacent lengths, …
Preview problemDecide whether a given set of side and angle measurements can form a triangle
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
For three proposed side lengths, sort them and focus on the longest. A nondegenerate triangle exists only when the two shorter lengths add to strictly more than the longest length; …
Preview problemRound and interpret a triangle measurement to explicit precision
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Keep the triangle calculation at full precision until the final reporting step. Identify the requested last place, inspect only the digit immediately to its right to round, and retain the …
Preview problemChoose the correct triangle law from the data
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Classify only the measurements actually given before deciding on a triangle law. Two known angles fix the third angle and eliminate SSA ambiguity; once a known side is matched with …
Preview problemConvert bearings into a solvable triangle
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Convert directional bearings into the triangle's interior angle before applying any triangle law. When two bearings share the same north reference, their angular separation is found from the appropriate difference; …
Preview problemSolve a fully specified triangulation model
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Model triangulation with the measured baseline and the two sight rays as one complete triangle. Find the remaining angle, then anchor the Law of Sines with the baseline and its …
Preview problemFind a resultant from non-right vector geometry
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Use the first vector as a reference axis and resolve the second into parallel and perpendicular components. Add the parallel contribution to the first magnitude while retaining the perpendicular contribution; …
Preview problemPreserve exact triangle-law results before rounding
Apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles.
Keep the Law of Cosines expression exact while matching the included angle with its opposite side. Use exact special-angle values, take the positive square root for a length, and simplify …
Preview problemFind the area of a triangle from two sides and the included angle
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Treat one given side as the base and view the sine component of the other side as the perpendicular height. Substituting that height into one-half base times height produces the …
Preview problemFind a missing side using the triangle area formula A = (1/2)ab sin(C)
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Start with the included-angle area formula and substitute the known area, side, and angle before rearranging. The known side's sine component supplies the perpendicular height, while the unknown side remains …
Preview problemFind all included angles from area and two sides
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
First compare the target area with the maximum possible area for the two fixed sides, which occurs when the included angle has sine one. If the target is attainable, isolate …
Preview problemFind a triangle’s area from two sides and the included angle by writing the altitude as side × sine
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Inventory the measurements before reaching for a formula. When a base and its corresponding perpendicular height are already supplied, one-half base times height uses the data directly; no trigonometric conversion …
Preview problemFind a triangle’s area from two sides and the included angle by expressing the altitude with sine
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Use either property boundary as the base and convert the other boundary into perpendicular height with the sine of their included angle. Substituting that height into one-half base times height …
Preview problemCompare triangle areas when two sides are fixed using A = 1/2ab sin(C)
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
With two side lengths fixed, the constant factor one-half times their product is the same for every configuration. Compare areas by comparing only the sine of each included angle; supplementary …
Preview problemWrite a triangle area formula from two sides and the included angle
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Start by locating the vertex where the two named sides meet; that vertex's angle is the included angle required by the area formula. Treat either side as the base and …
Preview problemEvaluate and interpret triangle area to stated precision
Derive the triangle area formula A=1/2ab sin(C) using an auxiliary altitude.
Preserve the exact trigonometric area model until calculator evaluation, and use degree mode for an angle measured in degrees. Carry full precision through the product, round only the final area …
Preview problemFactor a sum of squares over the complex numbers using conjugate imaginary factors
Extend polynomial identities to complex numbers for higher-degree polynomial work.
A real sum of squares becomes a difference of squares over the complex numbers because the square of an imaginary multiple is negative. Rewrite the constant square as the negative …
Preview problemFactor an even-power polynomial completely over the complex numbers using conjugate imaginary factors
Extend polynomial identities to complex numbers for higher-degree polynomial work.
When only even powers appear, treat the polynomial as a quadratic in a lower power such as x squared. Factor that quadratic structure first, substitute the original power back, and …
Preview problemFactor a cube polynomial over R and C and list roots
Extend polynomial identities to complex numbers for higher-degree polynomial work.
Use the sum- or difference-of-cubes identity to expose one real linear factor and a remaining quadratic. The real factor gives one root immediately; solve the quadratic without stopping just because …
Preview problemMultiply complex conjugate linear factors to write a real polynomial
Extend polynomial identities to complex numbers for higher-degree polynomial work.
Rewrite the two factors around their shared real center so the opposite imaginary parts form a conjugate pair. Their product is a difference of squares: the imaginary cross terms cancel, …
Preview problemVerify a complex zero of a real-coefficient polynomial using conjugate factors
Extend polynomial identities to complex numbers for higher-degree polynomial work.
To verify a proposed complex zero, substitute it for every occurrence of the variable and simplify the polynomial exactly. Reduce powers of the imaginary unit using i squared equals negative …
Preview problemFind all complex roots of an already factored polynomial, including roots from a quadratic that gives a conjugate imaginary pair
Extend polynomial identities to complex numbers for higher-degree polynomial work.
Apply the zero-product property to the given factorization and solve every factor independently. A quadratic equation of the form x squared equals a negative real number contributes both imaginary square …
Preview problemBuild the unique monic real polynomial from roots
Extend polynomial identities to complex numbers for higher-degree polynomial work.
Turn each listed root r into the factor x minus r. To keep coefficients real, pair every nonreal root with its conjugate before expanding; their product becomes a real quadratic …
Preview problem