Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Use congruent triangles to find a missing side or angle measure
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
The ordered congruence statement, rather than the drawing’s orientation, determines which parts correspond. We’ll decode the vertex mapping, map the requested side by both endpoints, invoke corresponding parts of congruent …
Preview problemUse similar triangles to find a missing side or angle measure
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
A scale factor is directional, so the words smaller-to-larger control the operation. We’ll match the corresponding sides, apply the factor in its stated direction, evaluate the product, and divide the …
Preview problemProve a geometric relationship after proving triangles congruent
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
CPCTC is a consequence used after triangle congruence has already been established. We’ll read the proved order, map the target side through its endpoints, apply corresponding-parts congruence to that pair, …
Preview problemProve a geometric relationship after proving triangles similar
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
A side-ratio conclusion from similarity must preserve one triangle direction across every fraction. We’ll decode the ordered vertex map, list all three corresponding side pairs, place first-triangle over second-triangle consistently, …
Preview problemSolve an indirect measurement problem with triangle similarity
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
Indirect measurement works because the person-shadow and building-shadow triangles share the same shape. We’ll justify AA from vertical objects, horizontal ground, and parallel sunlight, compare height to shadow in one …
Preview problemSolve a triangle design with validity, method, scale, and side fields
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
This design audit requires validity, classification, and scale to agree. We’ll test the side triple with the triangle inequality, compare all three corresponding lengths, apply the strongest matching criterion, compute …
Preview problemAudit evidence separately for congruence and similarity
Use triangle congruence and similarity criteria to solve problems and prove geometric relationships.
Evidence must be tested separately against the congruence and similarity criteria. We’ll classify the angle as non-included, examine the two-position ambiguity of SSA for non-right triangles, compare the data with …
Preview problemAssign separate opposite, adjacent-leg, and hypotenuse roles
Define trigonometric ratios for acute angles using right-triangle similarity.
Trig side roles are assigned relative to a reference angle, except that the hypotenuse is fixed by the right angle. We’ll locate that side first, find the leg across from …
Preview problemWrite the sine ratio for an acute angle in a right triangle
Define trigonometric ratios for acute angles using right-triangle similarity.
Sine is built from two side roles relative to the stated acute angle. We’ll recall opposite over hypotenuse, assign those roles before using the numbers, substitute in numerator-denominator order, simplify …
Preview problemWrite the cosine ratio for an angle in a right triangle
Define trigonometric ratios for acute angles using right-triangle similarity.
Cosine uses the adjacent leg, not merely any side touching the reference angle. We’ll identify the hypotenuse first, isolate the other touching side as the adjacent leg, substitute adjacent over …
Preview problemWrite the tangent ratio for an angle in a right triangle
Define trigonometric ratios for acute angles using right-triangle similarity.
Tangent compares the two legs relative to the reference angle, so order matters. We’ll identify the opposite and adjacent roles first, place them as opposite over adjacent, substitute their lengths, …
Preview problemCompute separate sine, cosine, and tangent fields
Define trigonometric ratios for acute angles using right-triangle similarity.
Similar right triangles preserve trig ratios because scaling multiplies both terms of each fraction equally. We’ll align opposite, adjacent, and hypotenuse roles, compute sine, cosine, and tangent from each triangle, …
Preview problemSolve for a missing side from a basic trigonometric equation
Define trigonometric ratios for acute angles using right-triangle similarity.
A trigonometric function value is a side ratio, not the angle measure itself. We’ll interpret the supplied sine equation, replace the special-angle sine with its exact ratio, solve the resulting …
Preview problemChoose the correct trigonometric ratio from the known and unknown sides
Define trigonometric ratios for acute angles using right-triangle similarity.
The correct trig function is the one whose definition contains both the known and requested side roles. We’ll inventory those roles, compare them against opposite-over-hypotenuse, adjacent-over-hypotenuse, and opposite-over-adjacent, and use …
Preview problemFind an acute angle from a special trigonometric ratio
Define trigonometric ratios for acute angles using right-triangle similarity.
Inverse sine turns a ratio into an angle, with the stated acute interval removing other possibilities. We’ll rewrite the decimal as an exact fraction, recognize its special-triangle relationship, apply inverse …
Preview problemChoose and interpret a contextual trig ratio
Define trigonometric ratios for acute angles using right-triangle similarity.
At the ground angle, a ramp’s rise and run are the opposite and adjacent legs. We’ll form tangent as rise over run, simplify while noting that the units cancel in …
Preview problemVerify both sine-cosine cofunction identities
Define trigonometric ratios for acute angles using right-triangle similarity.
Complementary acute angles view the same two legs with opposite and adjacent roles reversed. We’ll establish the ninety-degree sum, track that role swap while keeping the hypotenuse fixed, derive one …
Preview problemFind a complement and one matching cofunction identity
Explain and use the complementary-angle relationship between sine and cosine.
The numerical complement and the cofunction relationship come from the same right-triangle structure. We’ll subtract the known acute angle from ninety degrees, confirm the sum, track how opposite for one …
Preview problemRewrite sine of an acute angle as cosine of its complement
Explain and use the complementary-angle relationship between sine and cosine.
Rewriting a trig function with its cofunction requires changing to the complementary angle at the same time. We’ll compute ninety degrees minus the given angle, apply the sine-to-cosine identity, and …
Preview problemRewrite cosine of an acute angle as sine of its complement
Explain and use the complementary-angle relationship between sine and cosine.
A cosine-to-sine cofunction rewrite changes both the function name and the angle. We’ll calculate the acute complement, apply cosine of an angle equals sine of its complement, and check the …
Preview problemSolve a sine-cosine equation by using complementary acute angles
Explain and use the complementary-angle relationship between sine and cosine.
For acute angles, equal sine and cosine values signal complementary arguments rather than equal arguments. We’ll convert the cofunction equation into a ninety-degree angle sum, solve the resulting linear equation, …
Preview problemRewrite a sine expression as the cosine of its complementary angle
Explain and use the complementary-angle relationship between sine and cosine.
A sine-to-cosine rewrite uses the angle’s complement, not its supplement or the original angle. We’ll state the cofunction identity, subtract the given acute angle from ninety degrees, and place that …
Preview problemRewrite a cosine as the sine of its complementary angle
Explain and use the complementary-angle relationship between sine and cosine.
Switching from cosine to sine requires switching to the complementary acute angle as well. We’ll write the cofunction rule, calculate ninety degrees minus the given angle, and use that result …
Preview problemUse complementary angles to compare sine and cosine values
Explain and use the complementary-angle relationship between sine and cosine.
Different function names can represent the same ratio when their acute-angle arguments are complementary. We’ll add the two angles to test the ninety-degree condition, apply the sine-cosine cofunction identity, and …
Preview problemRelate a contextual complement through one cofunction equation
Explain and use the complementary-angle relationship between sine and cosine.
The two acute angles in the contextual right triangle are complements, and they view the same vertical leg differently. We’ll compute the missing angle from ninety degrees, track that leg …
Preview problemValidate or correct a sine-cosine cofunction statement
Explain and use the complementary-angle relationship between sine and cosine.
A cofunction statement should be checked structurally before its decimal or exact values are compared. We’ll compute the complement of the sine angle, verify that it matches the cosine angle, …
Preview problemFind the missing side of a right triangle by identifying the legs and hypotenuse
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
The Pythagorean setup begins by identifying which sides meet at the right angle and which side lies across from it. We’ll assign the leg and hypotenuse roles, square and add …
Preview problemUse sine to find a missing opposite side or hypotenuse in a right triangle
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Sine directly relates the opposite side and hypotenuse for the stated reference angle. We’ll assign those roles, write opposite over hypotenuse in the correct order, substitute the exact special-angle value, …
Preview problemUse cosine to find a missing adjacent side or hypotenuse in a right triangle
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Cosine is the direct bridge between the adjacent leg and hypotenuse. We’ll label those roles relative to the given angle, keep the hypotenuse in the denominator, insert the exact cosine …
Preview problemUse tangent to find a missing leg in a right triangle
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Because the known and unknown sides are the two legs, tangent connects them without introducing the hypotenuse. We’ll assign opposite and adjacent relative to the angle, write their ratio, use …
Preview problemFind a missing acute angle in a right triangle using inverse sine
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
To recover an angle from opposite and hypotenuse, we first build the sine ratio and then undo sine with its inverse. We’ll simplify the side fraction, apply inverse sine in …
Preview problemFind a missing acute angle in a right triangle using inverse cosine
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Adjacent and hypotenuse determine cosine, while inverse cosine returns the angle. We’ll form the ratio in the correct order, simplify it, apply inverse cosine on the acute interval, and substitute …
Preview problemUse inverse tangent to find a missing acute angle from the opposite and adjacent sides of a right triangle
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
The two leg lengths define tangent, and inverse tangent converts that ratio into the missing acute angle. We’ll place opposite over adjacent, simplify, apply the inverse function in degrees, and …
Preview problemSolve a right triangle into separate side and angle fields
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
A complete triangle solution must keep every named side tied to its angle role. We’ll find the complementary acute angle, use sine for the opposite leg and cosine for the …
Preview problemSolve one scalar elevation/depression target with units and precision
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
From the elevation angle at the shadow tip, height is opposite and shadow length is adjacent. We’ll form rise over run with tangent, simplify the ratio, apply inverse tangent in …
Preview problemSolve a structure problem with separate length and angle fields
Use trig ratios and the Pythagorean Theorem to solve right triangles in applied problems.
This structure problem has two targets that call for two connected methods. We’ll identify the ladder as the hypotenuse, use the Pythagorean theorem for the positive base distance, form a …
Preview problem