Higher Maths: trigonometry

Higher trigonometry asks for a set of formulae to be available instantly and applied without sign errors: the compound angle expansions, the double angle formulae, the Pythagorean identity, and the wave function that rewrites a sum of a sine and a cosine as a single shifted cosine.

Ten single-answer questions cover converting degrees to radians, the exact value of a standard angle, the expansion of sin(A + B), finding a double angle from a known sine, simplifying an expression with the Pythagorean identity, solving a cosine equation that has two solutions in the range, expressing $3\cos x + 4\sin x$ as a single wave, the maximum and minimum of a transformed sine graph, building the exact value of cos 75 degrees out of 45 and 30, and reading the period off a cosine graph.

The wrong options are the errors that actually appear: the sine and cosine compound formulae swapped, the factor of two dropped from a double angle, the amplitude found by adding the coefficients rather than by Pythagoras, the auxiliary angle taken from the wrong ratio, and the amplitude mistaken for the maximum of a graph that has been shifted vertically.

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• Convert between degrees and radians and give an answer in terms of pi
• Recall the exact values for the standard angles without confusing 30 with 60 degrees
• State and apply the compound angle formula for sine, distinguishing it from the cosine version
• Find a double angle from a known sine by first recovering the cosine
• Simplify an expression using the Pythagorean identity
• Solve a cosine equation and give every solution in the stated range
• Express a sum of a sine and a cosine as a single wave function, finding both the amplitude and the auxiliary angle
• Find the maximum and minimum of a sine graph that has been stretched and shifted
• Build an exact value from two standard angles using a compound angle formula
• Find the period of a trigonometric graph from the coefficient of x

Express $3\cos x^\circ + 4\sin x^\circ$ in the form $k\cos(x - a)^\circ$. — Here $k = \sqrt{3^2 + 4^2} = 5$, and equating coefficients gives $k\cos a = 3$ and $k\sin a = 4$, so $\tan a = \frac{4}{3}$ and $a \approx 53.1^\circ$.