> ## Content Index > Fetch the complete content index at: https://www.a-level-maths-tutor.com/llms.txt > Use this file to discover other available public pages before exploring further. # A-Level Maths | Trigonometry - URL: https://www.a-level-maths-tutor.com/a-level-maths-trigonometry/ - Published: 2026-09-18T19:11:00.000Z - Updated: 2026-09-18T19:11:00.000Z - Author: Jaisul Naik - Tags: Pure Maths Reciprocal Trigonometric Functions $$\\sec x = \\frac{1}{\\cos x} \\qquad \\cosec x = \\frac{1}{\\sin x} \\qquad \\cot x = \\frac{1}{\\tan x}$$ ### Question 1 Solve each of the following trigonometric equations. a) $\\sec\\theta = 4, \\quad 0 \\leq \\theta < 360°$ b) $3\\cot 2x - 1 = 4, \\quad 0 \\leq x < 180°$ c) $2\\cosec 2y = 10, \\quad 0 \\leq y < 2\\pi$ d) $8\\tan\\varphi = \\cot^2\\varphi, \\quad 0 \\leq \\varphi < 2\\pi$ e) $2\\sec\\theta = 3, \\quad 0 \\leq \\theta < 360°$ f) $\\cot 3x = \\dfrac{1}{4}, \\quad -90° \\leq x < 90°$ g) $5 - \\cosec 2y = -1, \\quad 0 \\leq y < 2\\pi$ h) $27\\sin^2\\varphi + 8\\cosec\\varphi = 0, \\quad 0 \\leq \\varphi < 2\\pi$ Show Answers a) $\\theta \\approx 75.5°,\\ 284.5°$ b) $x \\approx 15.5°,\\ 105.5°$ c) $y \\approx 0.10,\\ 1.47,\\ 3.24,\\ 4.61$ d) $\\varphi \\approx 0.46,\\ 3.61$ e) $\\theta \\approx 48.2°,\\ 311.8°$ f) $x \\approx -34.7°,\\ 25.3°,\\ 85.3°$ g) $y \\approx 0.0837,\\ 1.49,\\ 3.23,\\ 4.63$ h) $\\varphi \\approx 3.87,\\ 5.55$ Pythagorean Identities $$\\sec^2 x = 1 + \\tan^2 x \\qquad \\cosec^2 x = 1 + \\cot^2 x$$ ### Question 2 Solve each of the following equations. a) $2\\tan^2\\theta = 11\\sec\\theta - 7, \\quad 0 \\leq \\theta < 360°$ b) $4\\cot^2 x - 9\\cosec x + 6 = 0, \\quad 0 \\leq x < 360°$ c) $\\sec^2 y + \\tan y = 3, \\quad 0 \\leq y < 360°$ d) $2\\cosec^2\\varphi + \\cot^2\\varphi = 11, \\quad 0 \\leq \\varphi < 360°$ Show Answers a) $\\theta = 78.5°,\\ 281.5°$ b) $x = 30°,\\ 150°$ c) $y = 45°,\\ 225°,\\ y \\approx 116.6°,\\ 296.6°$ d) $\\varphi = 30°,\\ 150°,\\ 210°,\\ 330°$ Trigonometric Identities ### Question 3 Prove the validity of each of the following trigonometric identities. a) $\\dfrac{\\cot^2 x}{1+\\cot^2 x} \\equiv \\cos^2 x$ b) $\\dfrac{1}{\\sec x - \\tan x} + \\dfrac{1}{\\sec x + \\tan x} \\equiv 2\\sec x$ c) $\\dfrac{\\tan x \\sec x}{1 + \\tan^2 x} \\equiv \\sin x$ d) $\\dfrac{1}{\\sec x - \\tan x} - \\dfrac{1}{\\sec x + \\tan x} \\equiv 2\\tan x$ e) $\\dfrac{\\cot x \\cosec x}{1 + \\cot^2 x} \\equiv \\cos x$ Compound and Double Angle Equations $$\\sin(A \\pm B) = \\sin A\\cos B \\pm \\cos A\\sin B$$ $$\\cos(A \\pm B) = \\cos A\\cos B \\mp \\sin A\\sin B$$ $$\\sin 2A = 2\\sin A\\cos A$$ $$\\cos 2A = \\cos^2 A - \\sin^2 A = 1 - 2\\sin^2 A = 2\\cos^2 A - 1$$ ### Question 4 Solve each of the following trigonometric equations. a) $\\cos(\\theta + 30°) = \\sin\\theta, \\quad 0 \\leq \\theta < 360°$ b) $3\\cos(x + 30°) = \\sin(x - 60°), \\quad 0 \\leq x < 360°$ c) $\\sin(y - 30°) = \\sin(y + 45°), \\quad 0 \\leq y < 360°$ d) $\\sin(\\varphi + 30°) = \\cos(\\varphi - 45°), \\quad 0 \\leq \\varphi < 360°$ e) $\\cos(\\alpha - 60°) = \\cos(\\alpha - 45°), \\quad 0 \\leq \\alpha < 360°$ Show Answers a) $\\theta = 30°,\\ 210°$ b) $x = 60°,\\ 240°$ c) $y = 82.5°,\\ 262.5°$ d) $\\varphi = 52.5°,\\ 232.5°$ e) $\\alpha = 52.5°,\\ 232.5°$ Trigonometric Identities with Double Angles ### Question 5 Prove the validity of each of the following trigonometric identities. a) $\\sec\\theta\\cosec\\theta \\equiv 2\\cosec 2\\theta$ b) $\\tan\\theta + \\cot\\theta \\equiv 2\\cosec 2\\theta$ c) $\\dfrac{1 - \\cos 2x}{\\sin 2x} \\equiv \\tan x$ d) $\\dfrac{\\cos 2\\theta}{\\cos\\theta - \\sin\\theta} \\equiv \\cos\\theta + \\sin\\theta$ e) $\\dfrac{\\cos 2x}{\\sin x} + \\dfrac{\\sin 2x}{\\cos x} \\equiv \\cosec x$ f) $\\cot 2x + \\cosec 2x \\equiv \\cot x$ g) $\\cos 2x + \\tan x\\sin 2x \\equiv 1$ h) $\\dfrac{\\sin x}{1 - \\cos x} \\equiv \\cot\\dfrac{1}{2}x$ i) $\\sin 2\\theta \\equiv \\dfrac{2\\tan\\theta}{1 + \\tan^2\\theta}$ j) $\\dfrac{1}{\\cos\\theta - \\sin\\theta} - \\dfrac{1}{\\cos\\theta + \\sin\\theta} \\equiv 2\\sin\\theta\\sec 2\\theta$ R-Transformations $$R^2 = A^2 + B^2 \\qquad \\tan\\alpha = \\frac{B}{A}$$ ### Question 6 $f(x) \\equiv 2\\sin x + 2\\cos x,\\ x\\in\\mathbb{R}$ a) Express $f(x)$ in the form $R\\sin(x+\\alpha)$, $R>0$, $0<\\alpha<\\dfrac{\\pi}{2}$. b) State the minimum and maximum value of: i. $y = f\\!\\left(x - \\dfrac{\\pi}{2}\\right)$ ii. $y = 2f(x) + 1$ iii. $y = \\left\[f(x)\\right\]^2$ iv. $y = \\dfrac{10}{f(x) + 3\\sqrt{2}}$ Show Answers a) $f(x) \\equiv \\sqrt{8}\\sin\\!\\left(x + \\dfrac{\\pi}{4}\\right)$ b) i. $\\left\[-\\sqrt{8},\\ \\sqrt{8}\\right\]$ ii. $\\left\[-2\\sqrt{8}+1,\\ 2\\sqrt{8}+1\\right\]$ iii. $\\left\[0,\\ 8\\right\]$ iv. $\\left\[\\sqrt{2},\\ 5\\sqrt{2}\\right\]$ ### Question 7 $y \\equiv 2\\sqrt{2}\\cos x + 2\\sqrt{2}\\sin x,\\ x\\in\\mathbb{R}$ a) Express $y$ in the form $R\\sin(x+\\alpha)$, $R>0$, $0<\\alpha<\\dfrac{\\pi}{2}$. b) Solve the equation $y = 2$ for $0 < x < 2\\pi$. c) Write down the maximum value of $y$. d) Find the smallest positive value of $x$ for which this maximum value occurs. Show Answers a) $y \\equiv 4\\sin\\!\\left(x + \\dfrac{\\pi}{4}\\right)$ b) $x \\approx 1.83,\\ 6.02$ c) $4$ d) $x = \\dfrac{\\pi}{4}$ ### Question 8 a) Express $3\\cos 3x + 7\\sin 3x$ in the form $R\\cos(3x - \\alpha)$, where $R > 0$ and $0 < \\alpha < \\dfrac{1}{2}\\pi$. b) Give full details of a sequence of three transformations needed to transform the curve $y = \\cos x$ to the curve $y = 3\\cos 3x + 7\\sin 3x$. c) Determine the greatest value of $3\\cos 3x + 7\\sin 3x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs. d) Determine the least value of $3\\cos 3x + 7\\sin 3x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs. Show Answers a) $\\sqrt{58}\\cos(3x - 1.17)$ b) To be added c) Greatest value $\\sqrt{58}$, at $x \\approx 0.389$ d) Least value $-\\sqrt{58}$, at $x \\approx 1.44$ --- ### A-Level Maths Tutoring I offer one-to-one and small group A-Level Maths tutoring for students across the UK and internationally. 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