{"id":9171,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9171"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","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":"

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 θ:\n

      \t\"\"<\/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 θ=x, sin θ=y, tan θ=y\/x<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

        Example:<\/strong>
        \nFor a right triangle with sides 3, 4, 5 and θ opposite side 3: \"\"<\/p>\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 2π<\/li>\n
        • Cosine:<\/strong> Wave starting at 1, goes down to -1, back to 1, repeats every 2π<\/li>\n
        • Tangent:<\/strong> Repeats every πππ, vertical asymptotes at θ=π\/2+nπθ
          \n\t <\/li>\n<\/ul>\n

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

          \"\"<\/p>\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 <\/li>\n<\/ul>\n

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

              \n
            • Sin θ=1.5 → Impossible, as range is [-1,1]<\/li>\n
            • Tan π\/2 → 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 θ, cos θ → 2π<\/li>\n
                • tan θ → π<\/li>\n<\/ul>\n<\/li>\n
                • Symmetry:<\/strong>\n
                    \n
                  • Sine: Odd function → sin(-θ) = -sin θ<\/li>\n
                  • Cosine: Even function → cos(-θ) = cos θ<\/li>\n
                  • Tangent: Odd function → tan(-θ) = -tan θ<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

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

                    \"\"<\/p>\n

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

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

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

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

                      \"\"<\/p>\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°, 30°, 45°, 60°, 90°<\/li>\n
                      • Key values:\n

                        \t\"\"<\/li>\n<\/ul>\n

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

                        \"\"
                        \n <\/p>\n

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

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

                        \"\"<\/p>\n

                        \"\"<\/p>\n

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

                        \"\"<\/p>\n

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

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

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

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

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

                            \n
                          • θ = 30° → csc30° = 1\/(1\/2) = 2<\/li>\n
                          • cot45° = 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

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

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

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

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

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

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

                                    \n
                                  • Point at θ = 60°, r = 2 → x = 2 cos60° = 1, y = 2 sin60° = √3<\/li>\n<\/ul>\n

                                     <\/p>\n

                                     <\/p>\n

                                    \"\"<\/p>\n

                                     <\/p>\n

                                    \"\"<\/p>\n

                                    Example: <\/strong>Find the value of sin 31π3<\/em>\"\" .<\/p>\n

                                    Solution: – <\/strong>We know that values of sin x repeats after an interval of 2p. Therefore, sin<\/em>31π<\/em>3<\/em>=<\/em>sin10π+<\/em>π<\/em>3<\/em>=sin<\/em>π<\/em>3<\/em>=<\/em>3<\/em>2.<\/em>\"\"<\/p>\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 2π (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 π (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

                                       <\/p>\n","protected":false},"excerpt":{"rendered":"

                                      Unit: Trigonometric & Polar Functions Chapter: Core Trigonometric Functions: Sine, Cosine & Tangent 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[628],"tags":[],"class_list":["post-9171","post","type-post","status-publish","format-standard","hentry","category-ap-precalculus"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9171","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9171"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9171\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9171"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}