{"id":9947,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9947"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"core-trigonometric-functions-sine-cosine-tangent","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/core-trigonometric-functions-sine-cosine-tangent\/","title":{"rendered":"Core Trigonometric Functions: Sine, Cosine & Tangent"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Trigonometric & Polar Functions<\/strong><\/h2>\n

Chapter: <\/strong>Core Trigonometric Functions: Sine, Cosine & Tangent<\/strong><\/h3>\n

Reference: – Definition of Core Trigonometric Functions, Graphical Representation of Sine, Cosine & Tangent, Domain and Range, Periodicity & Symmetry, Amplitude, Frequency, and Phase Shift, Special Angles and Exact Values, Pythagorean Identities, Reciprocal Relationships, Applications of Core Functions, Connection to Polar Representation<\/em><\/p>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • Core Trigonometric Functions & Graphical Representation<\/li>\n
  • Domain, Range, Periodicity & Symmetry<\/li>\n
  • Pythagorean Identities & Reciprocal Relationships<\/li>\n
  • Application, properties & Polar functions<\/li>\n<\/ul>\n

    1. <\/strong>Definition of Core Trigonometric Functions<\/strong><\/p>\n

    Explanation:<\/u><\/strong><\/p>\n

    Trigonometric functions relate angles of a triangle to ratios of its sides. They are foundational in both geometry and calculus.<\/p>\n

      \n
    • Right Triangle Definition:<\/strong> For a right triangle with angle \u03b8:\n
      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n

    • Unit Circle Definition:<\/strong> Place the triangle inside a unit circle (radius = 1):\n
        \n
      • Any point on the circle is (x, y)<\/li>\n
      • cos \u03b8=x, sin \u03b8=y, tan \u03b8=y\/x<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

        Example:<\/strong>
        \nFor a right triangle with sides 3, 4, 5 and \u03b8 opposite side 3: <\/p>\n

        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        2. <\/strong>Graphical Representation of Sine, Cosine & Tangent<\/strong><\/p>\n

        Explanation:<\/u><\/strong><\/p>\n

          \n
        • Sine:<\/strong> Wave starting at 0, goes up to 1, down to -1, repeats every 2\u03c0<\/li>\n
        • Cosine:<\/strong> Wave starting at 1, goes down to -1, back to 1, repeats every 2\u03c0<\/li>\n
        • Tangent:<\/strong> Repeats every \u03c0\u03c0\u03c0, vertical asymptotes at \u03b8=\u03c0\/2+n\u03c0\u03b8
          \n\t\u00a0<\/li>\n<\/ul>\n

          Example Graph Observations:<\/strong>
          \n\u00a0<\/p>\n

          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          3. <\/strong>Domain and Range<\/strong><\/p>\n

          Explanation:<\/u><\/strong><\/p>\n

            \n
          • Domain:<\/strong> Set of all valid input angles<\/li>\n
          • Range:<\/strong> Set of all possible function values
            \n\t\u00a0<\/li>\n<\/ul>\n

            \"\"<\/p>\n

            Source: Kapdec.com<\/p>\n<\/div>\n

            \nExample:<\/strong><\/p>\n

              \n
            • Sin \u03b8=1.5 \u2192 Impossible, as range is [-1,1]<\/li>\n
            • Tan \u03c0\/2 \u2192 Undefined (vertical asymptote)<\/li>\n<\/ul>\n

              4. <\/strong>Periodicity & Symmetry<\/strong><\/p>\n

              Explanation:<\/u><\/strong><\/p>\n

                \n
              • Periodicity:<\/strong> Functions repeat after a certain interval\n
                  \n
                • sin \u03b8, cos \u03b8 \u2192 2\u03c0<\/li>\n
                • tan \u03b8 \u2192 \u03c0<\/li>\n<\/ul>\n<\/li>\n
                • Symmetry:<\/strong>\n
                    \n
                  • Sine: Odd function \u2192 sin(-\u03b8) = -sin \u03b8<\/li>\n
                  • Cosine: Even function \u2192 cos(-\u03b8) = cos \u03b8<\/li>\n
                  • Tangent: Odd function \u2192 tan(-\u03b8) = -tan \u03b8<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                    Example:<\/strong>
                    \n\u00a0<\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    5. <\/strong>Amplitude, Frequency, and Phase Shift<\/strong><\/p>\n

                    Explanation:<\/u><\/strong><\/p>\n

                      \n
                    • Amplitude:<\/strong> Height from midline \u2192 A in y = A sin x<\/li>\n
                    • Frequency:<\/strong> Number of cycles in 2\u03c0 \u2192 affects period<\/li>\n
                    • Phase Shift:<\/strong> Horizontal shift \u2192 y = sin (x – \u03c6)<\/li>\n<\/ul>\n

                      Example:<\/strong><\/p>\n

                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      6. <\/strong>Special Angles and Exact Values<\/strong><\/p>\n

                      Explanation:<\/u><\/strong>
                      \nAngles with exact trigonometric values that simplify calculations.<\/p>\n

                        \n
                      • Common angles: 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0<\/li>\n
                      • Key values:\n
                        \"\"<\/p>\n

                        Source: Kapdec.com<\/p>\n<\/div>\n<\/li>\n<\/ul>\n

                        Example:<\/strong><\/p>\n

                        \"\"<\/p>\n

                        Source: Kapdec.com<\/p>\n<\/div>\n

                        \n\u00a0<\/p>\n

                        7. <\/strong>Pythagorean Identities<\/strong><\/p>\n

                        Explanation:<\/u><\/strong>
                        \nRelate sine, cosine, and tangent algebraically:<\/p>\n

                        \"\"<\/p>\n

                        Source: Kapdec.com<\/p>\n<\/div>\n

                        \"\"<\/p>\n

                        Source: Kapdec.com<\/p>\n<\/div>\n

                        Example:<\/strong>
                        \n\u00a0<\/p>\n

                        \"\"<\/p>\n

                        Source: Kapdec.com<\/p>\n<\/div>\n

                        8. <\/strong>Reciprocal Relationships<\/strong><\/p>\n

                        Explanation:<\/u><\/strong><\/p>\n

                        Other trigonometric functions are reciprocals:<\/p>\n

                          \n
                        • csc \u03b8 = 1\/sin \u03b8<\/li>\n
                        • sec \u03b8 = 1\/cos \u03b8<\/li>\n
                        • cot \u03b8 = 1\/tan \u03b8<\/li>\n<\/ul>\n

                          Example:<\/strong><\/p>\n

                            \n
                          • \u03b8 = 30\u00b0 \u2192 csc30\u00b0 = 1\/(1\/2) = 2<\/li>\n
                          • cot45\u00b0 = 1\/1 = 1<\/li>\n<\/ul>\n

                            9. <\/strong>Applications of Core Functions<\/strong><\/p>\n

                            Explanation<\/u><\/strong>:<\/strong><\/p>\n

                            Trigonometric functions model periodic behavior in real life:<\/p>\n

                            \"\"<\/p>\n

                            Source: Kapdec.com<\/p>\n<\/div>\n

                            Example:<\/strong><\/p>\n

                              \n
                            • A wave of amplitude 2 units: y = 2 sin(\u03c0t) \u2192 oscillates between -2 and 2<\/li>\n<\/ul>\n

                              10. <\/strong>Connection to Polar Representation<\/strong><\/p>\n

                              Explanation<\/u><\/strong>:<\/strong>
                              \n\u00a0<\/p>\n

                                \n
                              • Polar coordinates (r, \u03b8): Point represented as distance from origin (r) and angle \u03b8<\/li>\n
                              • Conversion to Cartesian:\n
                                  \n
                                • x = r cos \u03b8<\/li>\n
                                • y = r sin \u03b8<\/li>\n<\/ul>\n<\/li>\n
                                • Unit circle links polar and trigonometry<\/li>\n<\/ul>\n

                                  Example:<\/strong><\/p>\n

                                    \n
                                  • Point at \u03b8 = 60\u00b0, r = 2 \u2192 x = 2 cos60\u00b0 = 1, y = 2 sin60\u00b0 = \u221a3<\/li>\n<\/ul>\n

                                    \u00a0<\/p>\n

                                    \u00a0<\/p>\n

                                    \"\"<\/p>\n

                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                    \u00a0<\/p>\n

                                    \"\"<\/p>\n

                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                    Example: <\/strong>Find the value of sin 31\u03c03<\/em><\/p>\n

                                    \"\"<\/p>\n

                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                    .<\/p>\n

                                    Solution: – <\/strong>We know that values of sin x repeats after an interval of 2p. Therefore, sin<\/em>31\u03c0<\/em>3<\/em>=<\/em>sin10\u03c0+<\/em>\u03c0<\/em>3<\/em>=sin<\/em>\u03c0<\/em>3<\/em>=<\/em>3<\/em>2.<\/em><\/p>\n

                                    \"\"<\/p>\n

                                    Source: Kapdec.com<\/p>\n<\/div>\n

                                    Key Points<\/u><\/strong><\/p>\n

                                      \n
                                    • Sine (sin), cosine (cos), and tangent (tan) are fundamental trigonometric functions used to model relationships between angles and sides of right triangles.<\/li>\n
                                    • The sine function (sin) represents the ratio of the length of the side opposite an angle to the length of the hypotenuse.<\/li>\n
                                    • The cosine function (cos) represents the ratio of the length of the adjacent side to the length of the hypotenuse.<\/li>\n
                                    • The tangent function (tan) represents the ratio of the length of the side opposite an angle to the length of the adjacent side.<\/li>\n
                                    • The sine and cosine functions have a periodicity of 2\u03c0 (or 360 degrees), meaning they repeat their values after every full revolution around the unit circle.<\/li>\n
                                    • The tangent function has a periodicity of \u03c0 (or 180 degrees), repeating its values after every half revolution.<\/li>\n
                                    • The sine and cosine functions are bounded between -1 and 1, while the tangent function can take any real value.<\/li>\n<\/ul>\n

                                      \u00a0<\/p>\n

                                      <\/p>\n