https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:trigonometric-functions-ncert-new/x0419e5b3b578592a:trigonometric-functions-of-sum-and-difference-of-two-angles/v/finding-ratios-for-4x-and-6x-of-angles 2025-03-01T15:52:37.13987578Z Finding ratios for 4x and 6x of angles We've derived double and triple angle formulae in our previous videos. In this video, we look at the strategy that helps us build on top of previously derived formulae. We use it to figure what the formulae for tan(4x) and cos(6x). Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finding ratios for 4x and 6x of angles We've derived double and triple angle formulae in our previous videos. In this video, we look at the strategy that helps us build on top of previously derived formulae. We use it to figure what the formulae for tan(4x) and cos(6x). https://cdn.kastatic.org/ka-youtube-converted/Ez46k02vFzM.mp4/Ez46k02vFzM.mp4 377 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/trigonometry-identity-review-fun 2021-05-05T14:53:24Z Trig angle addition identities Sal reviews 6 related trigonometric angle addition identities: sin(a+b), sin(a-c), cos(a+b), cos(a-b), cos(2a), and sin(2a). Sal Khan video https://cdn.kastatic.org/googleusercontent/q779WlSevH6c0FCkG6uJcgjnP-3oXG6igoQNkUMmaUC_iYs2ef5ZtpHgGVMziteKMOmz7gjlpqZzF16rq8z2hxo Trig angle addition identities Sal reviews 6 related trigonometric angle addition identities: sin(a+b), sin(a-c), cos(a+b), cos(a-b), cos(2a), and sin(2a). https://cdn.kastatic.org/ka-youtube-converted/a70-dYvDJZY.mp4/a70-dYvDJZY.mp4 667 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/cosine-addition-identity-example 2021-05-05T14:53:16Z Using the cosine angle addition identity Sal evaluates the cosine of the sum of 60° and another angle whose right triangle is given. To do this, he must use the cosine angle addition formula. Sal Khan video https://cdn.kastatic.org/googleusercontent/OirHRBIT_x63S6CU99h-fPyl2X_hDf7HZAM9tO9PaiIgSfaay7XSpcxQW-xYIn2jLljKXyz0qnd0vzmSZ1J7HWFVGA Using the cosine angle addition identity Sal evaluates the cosine of the sum of 60° and another angle whose right triangle is given. To do this, he must use the cosine angle addition formula. https://cdn.kastatic.org/ka-youtube-converted/h0SNEO25vIw.mp4/h0SNEO25vIw.mp4 314 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:trigonometric-functions-ncert-new/x0419e5b3b578592a:trigonometric-functions-of-sum-and-difference-of-two-angles/e/trigonometric-functions-of-sum-and-difference-of-angles 2025-02-27T12:31:05.94748833Z Trigonometric functions of sum and difference of angles Practice using the identities sin(x+y), cos(x+y) etc. exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/proof-angle-addition-sine 2021-05-05T14:53:16Z Proof of the sine angle addition identity Sal proves the identity sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y). Sal Khan video https://cdn.kastatic.org/googleusercontent/egZR3z6x8IGAPeU5EAdzsvyL4RnjWigy9B_OnXbOh9W2mw3jSIKSig9uvNCMNyx0M0UpGJrLARgaoIfUhhq7fZI Proof of the sine angle addition identity Sal proves the identity sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y). https://cdn.kastatic.org/ka-youtube-converted/R0EQg9vgbQw.mp4/R0EQg9vgbQw.mp4 506 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/proof-angle-addition-cosine 2021-05-05T14:53:19Z Proof of the cosine angle addition identity Sal proves the identity cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). Sal Khan video https://cdn.kastatic.org/googleusercontent/FasP1nWHi_ADiQthXb5kLHxdJw2ww_abdrkLF2zuHo7X1r1TOAfd37BLg1JOjGdEEn4-K8Bde1hl2aYF_vZzn6Zb Proof of the cosine angle addition identity Sal proves the identity cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y). https://cdn.kastatic.org/ka-youtube-converted/0VBQnR2h8XM.mp4/0VBQnR2h8XM.mp4 362 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:trig-id/v/trigonometry-unit-circle-symmetry 2021-05-05T14:53:18Z Sine & cosine identities: symmetry Sal finds several trigonometric identities for sine and cosine by considering horizontal and vertical symmetries of the unit circle. Sal Khan video https://cdn.kastatic.org/googleusercontent/hX3WT_E5jPWwlXRobWPxGJq4OYInNRchN78MjUoOWRLZ23yRk-N961XQLvHLPtx2NymQIUgGR3tt9gH8UrMZy3qZQA Sine & cosine identities: symmetry Sal finds several trigonometric identities for sine and cosine by considering horizontal and vertical symmetries of the unit circle. https://cdn.kastatic.org/ka-youtube-converted/tzQ7arA917E.mp4/tzQ7arA917E.mp4 478 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:trig-id/v/tan-symmetries-unit-circle 2021-05-05T14:53:19Z Tangent identities: symmetry Sal finds several trigonometric identities for tangent by considering horizontal and vertical symmetries of the unit circle. Sal Khan video https://cdn.kastatic.org/googleusercontent/aJtG2MsWehGm34mtnN3fAUg8BPOp0PGeiziyIRmQaK4Ebf6XpWlPiE20cqN1fkmi3chWHluIhcbuH_MBKkj_7IrQ Tangent identities: symmetry Sal finds several trigonometric identities for tangent by considering horizontal and vertical symmetries of the unit circle. https://cdn.kastatic.org/ka-youtube-converted/k_wJsio68D4.mp4/k_wJsio68D4.mp4 435 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:trig-id/v/trig-angle-rotations 2021-05-05T14:53:17Z Sine & cosine identities: periodicity Sal finds trigonometric identities for sine and cosine by considering angle rotations on the unit circle. Sal Khan video https://cdn.kastatic.org/googleusercontent/8R9ptji40BAkC_jOzCeGm6Fcnoe-QabEA_HoLfqCyjc0pTzHeTD_nCa92YdE8v1RV1B176jw6mZzX53nxrTkLjqx Sine & cosine identities: periodicity Sal finds trigonometric identities for sine and cosine by considering angle rotations on the unit circle. https://cdn.kastatic.org/ka-youtube-converted/h-TPSylHrvE.mp4/h-TPSylHrvE.mp4 373 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:trig-id/v/tan-periodicity 2021-05-05T14:53:17Z Tangent identities: periodicity Sal solves a problem by considering the periodicity of the tangent function. Sal Khan video https://cdn.kastatic.org/googleusercontent/5syk7G_FmY-hYDAt88Z5TlQSkzxwSrnumrebAKIXOM2UegBtXO_8qFabPmW7WxnYvRCzlPDcYN8JgVexBKAcx5c Tangent identities: periodicity Sal solves a problem by considering the periodicity of the tangent function. https://cdn.kastatic.org/ka-youtube-converted/C3HFAyigqoY.mp4/C3HFAyigqoY.mp4 242 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/double-angle-formula-for-cosine-example-c 2021-05-05T14:53:17Z Using the cosine double-angle identity The cosine double angle formula tells us that cos(2θ) is always equal to cos²θ-sin²θ. For example, cos(60) is equal to cos²(30)-sin²(30). We can use this identity to rewrite expressions or solve problems. See some examples in this video. Sal Khan video https://cdn.kastatic.org/googleusercontent/152bMm2bPz9qxWTKf3kD6x5ejlJKR-tLzP7uUkfHeN7OOT9OJRqTPVW19HlJJNGXMZ8DK5MLDrNHrPU91DCpPMs Using the cosine double-angle identity The cosine double angle formula tells us that cos(2θ) is always equal to cos²θ-sin²θ. For example, cos(60) is equal to cos²(30)-sin²(30). We can use this identity to rewrite expressions or solve problems. See some examples in this video. https://cdn.kastatic.org/ka-youtube-converted/D_smr0GBPvA.mp4/D_smr0GBPvA.mp4 208 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/trigonometry/trig-equations-and-identities/challenge-trig-problems/v/cosine-angle-addition-2 2020-07-26T05:38:46Z Trig challenge problem: cosine of angle-sum Sal is given cos(θ) and cos(φ) and he finds cos(θ+φ). To do that, he must first find sin(θ) and sin(φ) using the Pythagorean identity. Sal Khan video https://cdn.kastatic.org/googleusercontent/gt-svavuQuwt7iAWDnZV6UyGzNNMvbw25U-cxH5Gi0mS_GlRFN35JIfd91yrzf9VquOEKamykMfIkpy0E3aQpVrckA Trig challenge problem: cosine of angle-sum Sal is given cos(θ) and cos(φ) and he finds cos(θ+φ). To do that, he must first find sin(θ) and sin(φ) using the Pythagorean identity. https://cdn.kastatic.org/ka-youtube-converted/XGpEHj43kcc.mp4/XGpEHj43kcc.mp4 434 Trigonometric functions of sum and difference of two angles https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:trigonometric-functions-ncert-new/x0419e5b3b578592a:trigonometric-functions-of-sum-and-difference-of-two-angles/e/sum-and-difference-of-trigonometric-functions 2026-04-07T23:36:17.218592739Z Sum and difference of trigonometric functions Test students' memory on the 4 identities - cos x + cos y, cos x - cos y, sin x + sin y, sin x - sin y. exercise https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:trigonometric-functions-ncert-new/x0419e5b3b578592a:trigonometric-functions-of-sum-and-difference-of-two-angles/e/evaluate-trigonometric-expressions-intermediate 2025-03-01T15:52:37.13987578Z Evaluate trigonometric expressions (intermediate) reduce given expressions into a simpler formula utilising various identities. exercise